Recent zbMATH articles in MSC 60https://zbmath.org/atom/cc/602024-03-13T18:33:02.981707ZUnknown authorWerkzeugDynamic probability and the problem of initial conditionshttps://zbmath.org/1528.000052024-03-13T18:33:02.981707Z"Strevens, Michael"https://zbmath.org/authors/?q=ai:strevens.michaelSummary: Dynamic approaches to understanding probability in the non-fundamental sciences turn on certain properties of physical processes that are apt to produce ``probabilistically patterned'' outcomes. The dynamic properties on their own, however, seem not quite sufficient to explain the patterns; in addition, some sort of assumption about initial conditions must be made, an assumption that itself typically takes a probabilistic form. How should such a posit be understood? That is the problem of initial conditions. Reichenbach, in his doctoral dissertation, floated a Kantian solution to the problem. In this paper I provide a Reichenbachian alternative.Invariant measures concentrated on countable structureshttps://zbmath.org/1528.031532024-03-13T18:33:02.981707Z"Ackerman, Nathanael"https://zbmath.org/authors/?q=ai:ackerman.nathanael-leedom"Freer, Cameron"https://zbmath.org/authors/?q=ai:freer.cameron-e"Patel, Rehana"https://zbmath.org/authors/?q=ai:patel.rehanaSummary: Let \(L\) be a countable language. We say that a countable infinite \(L\)-structure \(\mathcal{M}\) admits an invariant measure when there is a probability measure on the space of \(L\)-structures with the same underlying set as \(\mathcal{M}\) that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of \(\mathcal{M}\). We show that \(\mathcal{M}\) admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in \(\Aut(\mathcal{M})\) of an arbitrary finite tuple of \(\mathcal{M}\) fixes no additional points. When \(\mathcal{M}\) is a Fraïssé limit in a relational language, this amounts to requiring that the age of \(\mathcal{M}\) have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.Brooks' theorem for measurable coloringshttps://zbmath.org/1528.031962024-03-13T18:33:02.981707Z"Conley, Clinton T."https://zbmath.org/authors/?q=ai:conley.clinton-taylor"Marks, Andrew S."https://zbmath.org/authors/?q=ai:marks.andrew-s"Tucker-Drob, Robin D."https://zbmath.org/authors/?q=ai:tucker-drob.robin-dSummary: We generalize Brooks' theorem to show that if \(G\) is a Borel graph on a standard Borel space \(X\) of degree bounded by \(d\geqslant 3\) which contains no \((d+1)\)-cliques, then \(G\) admits a \(\mu\)-measurable \(d\)-coloring with respect to any Borel probability measure \(\mu\) on \(X\), and a Baire measurable \(d\)-coloring with respect to any compatible Polish topology on \(X\). The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of IID \(d\)-colorings of Cayley graphs of degree \(d\), except in two exceptional cases.Sharp Poincaré and log-Sobolev inequalities for the switch chain on regular bipartite graphshttps://zbmath.org/1528.050602024-03-13T18:33:02.981707Z"Tikhomirov, Konstantin"https://zbmath.org/authors/?q=ai:tikhomirov.konstantin-e"Youssef, Pierre"https://zbmath.org/authors/?q=ai:youssef.pierreSummary: Consider the switch chain on the set of \(d\)-regular bipartite graphs on \(n\) vertices with \(3\le d\le n^c\), for a small universal constant \(c>0\). We prove that the chain satisfies a Poincaré inequality with a constant of order \(O(nd)\); moreover, when \(d\) is fixed, we establish a log-Sobolev inequality for the chain with a constant of order \(O_d(n\log n)\). We show that both results are optimal. The Poincaré inequality implies that in the regime \(3\le d\le n^c\) the mixing time of the switch chain is at most \(O\big ((nd)^2 \log (nd)\big)\), improving on the previously known bound \(O\big ((nd)^{13} \log (nd)\big )\) due to \textit{R. Kannan} et al. [Random Struct. Algorithms 14, No. 4, 293--308 (1999; Zbl 0933.05145)] and \(O\big (n^7d^{18} \log (nd)\big )\) obtained by \textit{M. Dyer} et al. [Discrete Math. 344, No. 11, Article ID 112566, 14 p. (2021; Zbl 1472.05117)]. The log-Sobolev inequality that we establish for constant \(d\) implies a bound \(O(n\log^2 n)\) on the mixing time of the chain which, up to the \(\log n\) factor, captures a conjectured optimal bound. Our proof strategy relies on building, for any fixed function on the set of \(d\)-regular bipartite simple graphs, an appropriate extension to a function on the set of multigraphs given by the configuration model. We then establish a comparison procedure with the well studied random transposition model in order to obtain the corresponding functional inequalities. While our method falls into a rich class of comparison techniques for Markov chains on different state spaces, the crucial feature of the method -- dealing with chains with a large distortion between their stationary measures -- is a novel addition to the theory.Local limits of spatial inhomogeneous random graphshttps://zbmath.org/1528.050612024-03-13T18:33:02.981707Z"Van Der Hofstad, Remco"https://zbmath.org/authors/?q=ai:van-der-hofstad.remco-w"Van Der Hoorn, Pim"https://zbmath.org/authors/?q=ai:van-der-hoorn.pim"Maitra, Neeladri"https://zbmath.org/authors/?q=ai:maitra.neeladriSummary: Consider a set of \(n\) vertices, where each vertex has a location in \(\mathbb{R}^d\) that is sampled uniformly from the unit cube in \(\mathbb{R}^d\), and a weight associated to it. Construct a random graph by placing edges independently for each vertex pair with a probability that is a function of the distance between the locations and the vertex weights.
Under appropriate integrability assumptions on the edge probabilities that imply sparseness of the model, after appropriately blowing up the locations, we prove that the local limit of this random graph sequence is the (countably) infinite random graph on \(\mathbb{R}^d\) with vertex locations given by a homogeneous Poisson point process, having weights which are independent and identically distributed copies of limiting vertex weights. Our set-up covers many sparse geometric random graph models from the literature, including geometric inhomogeneous random graphs (GIRGs), hyperbolic random graphs, continuum scale-free percolation, and weight-dependent random connection models.
We prove that the limiting degree distribution is mixed Poisson and the typical degree sequence is uniformly integrable, and we obtain convergence results on various measures of clustering in our graphs as a consequence of local convergence. Finally, as a byproduct of our argument, we prove a doubly logarithmic lower bound on typical distances in this general setting.Growth network models with random number of attached linkshttps://zbmath.org/1528.050622024-03-13T18:33:02.981707Z"Sidorov, Sergei"https://zbmath.org/authors/?q=ai:sidorov.sergei-vladimirovich|sidorov.sergei-petrovich"Mironov, Sergei"https://zbmath.org/authors/?q=ai:mironov.sergei-grigorevich|mironov.sergei-vladimirovichSummary: In the process of many real networks' growth, each new node joins a few existing nodes, the number of which is unknown in advance. However, classical network growth models assume that the number of nodes to which each newborn node links is constant. In this regard, the main research question of this paper is as follows: how do structural properties of networks generated in accordance with growth models with random number of attached links, change (in comparison with networks with a constant number of attached links)? In this paper, we restrict our analysis to the two growth network models: the Barabási-Albert (BA) model and the triadic closure (TC) model. However, both in the BA and in the TC models the number of links attached to each new node of the network is constant on every iteration, which is not quite realistic. In this paper we extend both the BA and the TC models by allowing the number of newly added links to be random, under some mild assumptions on its distribution law. We examine the geometric properties of networks generated by the proposed models analytically and empirically to show that their properties differ from those of the classical BA and TC models. In particular, while the degree distributions of the generated networks follow the power law, their exponents vary depending on the thickness of the tail of the distribution that generates the number of attached links, and may differ significantly from the value of the corresponding exponent \(\gamma = -3\) for the networks generated by the classical BA and TC model. The `random' versions of models have a useful feature: they are capable of creating new nodes with a high degree at any iteration. This greatly distinguishes them from classical models, in which the nodes that appeared in the early iterations gain a significant advantage. The property of the proposed model, associated with the possibility of late arrivals to receive a random number of neighbors (which can be arbitrarily large), seems to be capable to simulate the temporal behavior of real networks in a more realistic way.Bimonoidal structure of probability monadshttps://zbmath.org/1528.180152024-03-13T18:33:02.981707Z"Fritz, Tobias"https://zbmath.org/authors/?q=ai:fritz.tobias"Perrone, Paolo"https://zbmath.org/authors/?q=ai:perrone.paoloSummary: We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimonoidal structure is then the correct requirement for the more general case.
We explain the theory and the operational interpretation, with the help of the graphical calculus for monoidal categories. We give a definition of stochastic independence based on the bimonoidal structure, compatible with the intuition and with other approaches in the literature for cartesian monoidal categories. We then show as an example that the Kantorovich monad on the category of complete metric spaces is a bimonoidal monad for a non-cartesian monoidal structure.
For the entire collection see [Zbl 1411.68020].Weighted cogrowth formula for free groupshttps://zbmath.org/1528.200302024-03-13T18:33:02.981707Z"Jaerisch, Johannes"https://zbmath.org/authors/?q=ai:jaerisch.johannes"Matsuzaki, Katsuhiko"https://zbmath.org/authors/?q=ai:matsuzaki.katsuhikoSummary: We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group \(\text{Cay}(F_n)\) by an arbitrary subgroup \(G\) of \(F_n\). Our main result, which generalizes Grigorchuk's cogrowth formula to variable edge lengths, provides a formula relating the bottom of the spectrum of weighted Laplacian on \(G \backslash \text{Cay}(F_n)\) to the Poincaré exponent of \(G\). Our main tool is the Patterson-Sullivan theory for Cayley graphs with variable edge lengths.The Buffon's needle problem for random planar disk-like Cantor setshttps://zbmath.org/1528.280132024-03-13T18:33:02.981707Z"Vardakis, Dimitris"https://zbmath.org/authors/?q=ai:vardakis.dimitris"Volberg, Alexander"https://zbmath.org/authors/?q=ai:volberg.alexander-lSummary: We consider a model of randomness for self-similar Cantor sets of finite and positive 1-Hausdorff measure. We find the sharp rate of decay of the probability that a Buffon needle lands \(\delta\)-close to a Cantor set of this particular randomness. Two quite different models of randomness for Cantor sets, by \textit{Y. Peres} and \textit{B. Solomyak} [Pac. J. Math. 204, No. 2, 473--496 (2002; Zbl 1046.28006)], and by \textit{S. Zhang} [Rev. Mat. Iberoam. 36, No. 2, 537--548 (2020; Zbl 1437.28018)], appear to have the same order of decay for the Buffon needle probability: \(\frac{c}{\log \frac{1}{\delta}}\). In this note, we prove the same rate of decay for a third model of randomness, which asserts a vague feeling that any ``reasonable'' random Cantor set of positive and finite length will have Favard length of order \(\frac{c}{\log \frac{1}{\delta}}\) for its \(\delta\)-neighbourhood. The estimate from below was obtained long ago by \textit{P. Mattila} [Ann. Acad. Sci. Fenn., Ser. A I, Math. 1, 227--244 (1975; Zbl 0348.28019)].Box-counting dimension and differentiability of box-like statistically self-affine functionshttps://zbmath.org/1528.280142024-03-13T18:33:02.981707Z"Allaart, Pieter"https://zbmath.org/authors/?q=ai:allaart.pieter-c"Jones, Taylor"https://zbmath.org/authors/?q=ai:jones.taylorSummary: We consider a class of ``box-like'' statistically self-affine functions, and compute the almost-sure box-counting dimension of their graphs. Furthermore, we consider the differentiability of our functions, and prove that, depending on an explicitly computable functional of the model, they are almost surely either differentiable almost everywhere or non-differentiable almost everywhere.Iterated function systems based on the degree of nondensifiabilityhttps://zbmath.org/1528.280202024-03-13T18:33:02.981707Z"García, Gonzalo"https://zbmath.org/authors/?q=ai:garcia.gonzalo-a"Mora, Gaspar"https://zbmath.org/authors/?q=ai:mora.gasparSummary: In the present paper we introduce the concept of iterated function systems (IFS) having at least one \(\phi \)-condensing mapping which belongs to the finite set of self-mappings that define the IFS. It is shown the existence of an invariant for those IFS. Whenever all the self-mappings are \(\phi\)-condensing we prove that the invariant set is compact. We propose some applications of those IFS having \(\phi\)-condensing self-mappings to the superposition operator defined on the Banach space \(\mathcal{C} ( [ 0 , 1 ] )\).Kähler-Einstein metrics and Archimedean zeta functionshttps://zbmath.org/1528.320372024-03-13T18:33:02.981707Z"Berman, Robert J."https://zbmath.org/authors/?q=ai:berman.robert-jSummary: While the existence of a unique Kähler-Einstein metric on a canonically polarized manifold \(X\) was established by Aubin and Yau already in the 70s, there are only a few explicit formulas available. In a previous work, a probabilistic construction of the Kähler-Einstein metric was introduced -- involving canonical random point processes on \(X\) -- which yields canonical approximations of the Kähler-Einstein metric, expressed as explicit period integrals over a large number of products of \(X\). Here it is shown that the conjectural extension to the case when \(X\) is a Fano variety suggests a zero-free property of the Archimedean zeta functions defined by the partition functions of the probabilistic model. A weaker zero-free property is also shown to be relevant for the Calabi-Yau equation. The convergence in the case of log Fano curves is settled, exploiting relations to the complex Selberg integral in the orbifold case. Some intriguing relations to the zero-free property of the local automorphic \(L\)-functions appearing in the Langlands program and arithmetic geometry are also pointed out. These relations also suggest a natural \(p\)-adic extension of the probabilistic approach.
For the entire collection see [Zbl 1519.00033].\(q\)-Pearson pair and moments in \(q\)-deformed ensembleshttps://zbmath.org/1528.330232024-03-13T18:33:02.981707Z"Forrester, Peter J."https://zbmath.org/authors/?q=ai:forrester.peter-j"Li, Shi-Hao"https://zbmath.org/authors/?q=ai:li.shihao"Shen, Bo-Jian"https://zbmath.org/authors/?q=ai:shen.bo-jian"Yu, Guo-Fu"https://zbmath.org/authors/?q=ai:yu.guofuSummary: The generalisation of continuous orthogonal polynomial ensembles from random matrix theory to the \(q\)-lattice setting is considered. We take up the task of initiating a systematic study of the corresponding moments of the density from two complementary viewpoints. The first requires knowledge of the ensemble average with respect to a general Schur polynomial, from which the spectral moments follow as a corollary. In the case of little \(q\)-Laguerre weight, a particular \({}_3\phi_2\) basic hypergeometric polynomial is used to express density moments. The second approach is to study the \(q\)-Laplace transform of the un-normalised measure. Using integrability properties associated with the \(q\)-Pearson equation for the \(q\)-classical weights, a fourth-order \(q\)-difference equation is obtained, generalising a result of Ledoux in the continuous classical cases.Central exponents of linear stochastic differential-algebraic equations of index 1https://zbmath.org/1528.340162024-03-13T18:33:02.981707Z"Nguyen Thi The"https://zbmath.org/authors/?q=ai:nguyen-thi-the.Summary: Our aim in this paper is to establish a concept of central exponents of a stochastic differential-algebraic equation (SDAE) of index 1. For this purpose, we introduce the inherent regular stochastic differential equation (SDE) associated with a SDAE of index 1. Then, the central exponents are defined samplewise via the induced two-parameter stochastic flow generated by the inherent regular SDE. We prove that the central exponents are nonrandom and greater than or equal to the Lyapunov exponents.Results on approximate controllability for fractional stochastic delay differential systems of order \(r \in (1, 2)\)https://zbmath.org/1528.340612024-03-13T18:33:02.981707Z"Dineshkumar, C."https://zbmath.org/authors/?q=ai:dineshkumar.chendrayan"Vijayakumar, V."https://zbmath.org/authors/?q=ai:vijayakumar.velusamy"Udhayakumar, R."https://zbmath.org/authors/?q=ai:udhayakumar.r"Nisar, Kottakkaran Sooppy"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaran"Shukla, Anurag"https://zbmath.org/authors/?q=ai:shukla.anuragSummary: In this paper, we deal with the approximate controllability of fractional stochastic delay differential inclusions of order \(r \in (1, 2)\). By using fractional calculus, stochastic analysis, the theory of cosine family and Dhage fixed point techniques, a new set of necessary and sufficient conditions are formulated which guarantees the approximate controllability of the nonlinear fractional stochastic system. In particular, the results are established with the assumption that the associated linear part of the system is approximately controllable. Further, the result is extended to obtain the conditions for the solvability of controllability results for fractional inclusions with nonlocal conditions. Finally, an example is presented to illustrate the theory of the obtained result.Rigorous derivation of the Fick cross-diffusion system from the multi-species Boltzmann equation in the diffusive scalinghttps://zbmath.org/1528.350822024-03-13T18:33:02.981707Z"Briant, Marc"https://zbmath.org/authors/?q=ai:briant.marc"Grec, Bérénice"https://zbmath.org/authors/?q=ai:grec.bereniceSummary: We present the arising of the Fick cross-diffusion system of equations for fluid mixtures from the multi-species Boltzmann equation in a rigorous manner in Sobolev spaces. To this end, we formally show that, in a diffusive scaling, the hydrodynamical limit of the kinetic system is the Fick model supplemented with a closure relation and we give explicit formulae for the macroscopic diffusion coefficients from the Boltzmann collision operator. Then, we provide a perturbative Cauchy theory in Sobolev spaces for the constructed Fick system, which turns out to be a dilated parabolic equation. We finally prove the stability of the system in the Boltzmann equation, ensuring a rigorous derivation between the two models.Long time behavior of stochastic NLS with a small multiplicative noisehttps://zbmath.org/1528.351582024-03-13T18:33:02.981707Z"Fan, Chenjie"https://zbmath.org/authors/?q=ai:fan.chenjie"Xu, Weijun"https://zbmath.org/authors/?q=ai:xu.weijun"Zhao, Zehua"https://zbmath.org/authors/?q=ai:zhao.zehuaSummary: We prove the global space-time bound for the defocusing mass critical nonlinear Schrödinger equation on \(\mathbb{R}^3\) perturbed by a small multiplicative noise. The associated scattering behavior is also obtained. In addition to techniques from [\textit{C. Fan} and \textit{W. Xu}, Anal. PDE 14, No. 8, 2561--2594 (2021; Zbl 1490.35412); \textit{C. Fan} and \textit{Z. Zhao}, ``On long time behavior for stochastic nonlinear Schrödinger equations with a multiplicative noise'', Preprint, \url{arXiv:2010.11045}], the main new ingredients are the decomposition of the solution tailored for the bootstrap argument in this problem, and the incorporation of local smoothing norms to close the argument. We also prove the global space-time Strichartz estimate for the linear stochastic equation. It is a toy model of our nonlinear problem, but the bound itself is new and of its own interest. Furthermore, the proof we give for the linear model is more direct, and also illustrates the proof strategy for the nonlinear problem.Nonlinear fractional damped wave equation on compact Lie groupshttps://zbmath.org/1528.352182024-03-13T18:33:02.981707Z"Dasgupta, Aparajita"https://zbmath.org/authors/?q=ai:dasgupta.aparajita"Kumar, Vishvesh"https://zbmath.org/authors/?q=ai:kumar.vishvesh"Mondal, Shyam Swarup"https://zbmath.org/authors/?q=ai:mondal.shyam-swarupIn this paper, the authors considered the initial value fractional damped wave equation on G, a compact Lie group, with power-type nonlinearity. By making use of Fourier analysis on compact Lie groups, they proved a local in-time existence result in the energy space for the fractional damped wave equation on G. Moreover, a finite time blow-up result was established under certain conditions on the initial data. The authors also studied one class of fractional wave equations with lower order terms, i.e., damping and mass with the same power type nonlinearity on compact Lie groups, and proved the global in-time existence of small data solutions in the energy evolution space.
Reviewer: Changxing Miao (Beijing)Regularity of transition densities and ergodicity for affine jump-diffusionshttps://zbmath.org/1528.370012024-03-13T18:33:02.981707Z"Friesen, Martin"https://zbmath.org/authors/?q=ai:friesen.martin"Jin, Peng"https://zbmath.org/authors/?q=ai:jin.peng"Kremer, Jonas"https://zbmath.org/authors/?q=ai:kremer.jonas"Rüdiger, Barbara"https://zbmath.org/authors/?q=ai:rudiger.barbaraSummary: This paper studies the transition density and exponential ergodicity for affine processes on the canonical state space \(\mathbb{R}_{\geq 0}^m \times \mathbb{R}^n\). Under a Hörmander-type condition for diffusion components as well as a boundary nonattainment condition, we derive the existence and regularity of the transition densities and then prove the strong Feller property of the associated semigroup. Moreover, we also show that, under an additional subcriticality condition on the drift, the corresponding affine process is exponentially ergodic in the total variation distance.
{{\copyright} 2022 Wiley-VCH GmbH.}Product type potential on the one-dimensional lattice systems: selection of maximizing probability and a large deviation principlehttps://zbmath.org/1528.370072024-03-13T18:33:02.981707Z"Mohr, J."https://zbmath.org/authors/?q=ai:mohr.jurgen|mohr.john-w|mohr.jonathan|mohr.james|mohr.joanaOne of the main results that the author proves is the existence of an equilibrium probability measure for a one-dimensional lattice system. More precisely, consider the set of functions \(\Omega=[0,1]^{\mathbb{N}}\) from the natural numbers to the closed interval \([0,1]\). Here \(\mathbb{N}=\{1,2,3,\dots\}\). A real-valued function \(f:\Omega\to\mathbb{R}\) is assumed to take the form \(f(x_1,x_2,\dots)=\sum^\infty_{j=1}f_j(x_j)\), where each real-valued function \(f_j:[0,1]\to\mathbb{R}\) is Lipschitz continuous with Lipschitz constant being less than \(\frac{1}{2^j}\) and periodic, \(f_j(0)=f_j(1)\). Consider the real-valued function defined as \(F(a)=f(a,a,a,\dots)\) with some differentiability conditions (please see the conditions in Theorem 7). This is assumed to achieve maximum value at one point \(a_1\). Define the probability measure \[\tilde{\mu}_{0,\beta}=\frac{e^{\beta F(a)}}{\int_0^1e^{\beta F(a)}da}da\] on the closed interval \([0,1]\) and the probability measure \(\tilde{\mu}_\beta=\bigotimes^\infty_{i=1}\tilde{\mu}_{0,\beta}\) on \(\Omega\). Then the author proves the following two results:
(1) There holds:
\[\lim\limits_{\beta\to+\infty}\tilde{\mu}_\beta\bigoplus^\infty_{i=1}\delta_{a_1};\]
(2) There holds:
\[\sum^\infty_{j=1}f_j(a_1)=\max\limits_{\mu\in\mathcal{M}_\sigma}\int_{\Omega}fd\mu,\]
where \(\mathcal{M}_\sigma\) is the set of invariant probability measures with respect to \(\sigma:\Omega\to\Omega\), \(\sigma(x_1,x_2,\dots)=(x_2,x_3,\dots)\).
The ideas of the proofs are as follows. Regarding statement (1), since the function \(F(a)\) achieves its maximum at one point \(a_1\) as \(\beta\to+\infty\), the support for the probability measure \(\tilde{\mu}_{0,\beta}\) gets concentrated near \(a_1\), and in the limit \(\beta\to+\infty\), one gets the Dirac delta function \(\delta_{a_1}\). Regarding statement (2), the assumptions on the function \(f(x_1,x_2,\dots)\) and statement (1) imply the conditions of Theorem 5 in [\textit{A. O. Lopes} et al., Ergodic Theory Dyn. Syst. 35, No. 6, 1925--1961 (2015; Zbl 1352.37090)], that in turn implies statement (2).
Reviewer: Haru Pinson (Tucson)On some aspects of the response to stochastic and deterministic forcingshttps://zbmath.org/1528.370422024-03-13T18:33:02.981707Z"Santos Gutiérrez, Manuel"https://zbmath.org/authors/?q=ai:santos-gutierrez.manuel"Lucarini, Valerio"https://zbmath.org/authors/?q=ai:lucarini.valerioSummary: The perturbation theory of operator semigroups is used to derive response formulas for a variety of combinations of acting forcings and reference background dynamics. In the case of background stochastic dynamics, we decompose the response formulas using the Koopman operator generator eigenfunctions and the corresponding eigenvalues, thus providing a functional basis towards identifying relaxation timescales and modes and towards relating forced and natural fluctuations in physically relevant systems. To leading order, linear response gives the correction to expectation values due to extra deterministic forcings acting on either stochastic or chaotic dynamical systems. When considering the impact of weak noise, the response is linear in the intensity of the (extra) noise for background stochastic dynamics, while the second order response given the leading order correction when the reference dynamics is chaotic. In this latter case we clarify that previously published diverging results can be brought to common ground when a suitable interpretation -- Stratonovich vs Itô -- of the noise is given. Finally, the response of two-point correlations to perturbations is studied through the resolvent formalism via a perturbative approach. Our results allow, among other things, to estimate how the correlations of a chaotic dynamical system changes as a results of adding stochastic forcing.Existence and stability results of stochastic differential equations with non-instantaneous impulse and Poisson jumpshttps://zbmath.org/1528.370442024-03-13T18:33:02.981707Z"Varshini, S."https://zbmath.org/authors/?q=ai:varshini.s"Banupriya, K."https://zbmath.org/authors/?q=ai:banupriya.k"Ramkumar, K."https://zbmath.org/authors/?q=ai:ramkumar.kasinathan"Ravikumar, K."https://zbmath.org/authors/?q=ai:ravikumar.kasinathanSummary: This paper focuses on a new class of non-instantaneous impulsive stochastic differential equations generated by mixed fractional Brownian motion with poisson jump in real separable Hilbert space. A set of sufficient conditions are generated based on the stochastic analysis technique, analytic semigroup theory of linear operators, fractional power of operators, and fixed point theory to obtain existence and uniqueness results of mild solutions for the considered system. Furthermore, the asymptotic behaviour of the system is investigated. Finally, an example is proposed to validate the obtained results.Integrable equations associated with the finite-temperature deformation of the discrete Bessel point processhttps://zbmath.org/1528.370582024-03-13T18:33:02.981707Z"Cafasso, Mattia"https://zbmath.org/authors/?q=ai:cafasso.mattia"Ruzza, Giulio"https://zbmath.org/authors/?q=ai:ruzza.giulioSummary: We study the finite-temperature deformation of the discrete Bessel point process. We show that its largest particle distribution satisfies a reduction of the 2D Toda equation, as well as a discrete version of the integro-differential Painlevé II equation of \textit{G. Amir} et al. [Commun. Pure Appl. Math. 64, No. 4, 466--537 (2011; Zbl 1222.82070)], and we compute initial conditions for the Poissonization parameter equal to 0. As proved by \textit{D. Betea} and \textit{J. Bouttier} [Math. Phys. Anal. Geom. 22, No. 1, Paper No. 3, 47 p. (2019; Zbl 1409.82010)], in a suitable continuum limit the last particle distribution converges to that of the finite-temperature Airy point process. We show that the reduction of the 2D Toda equation reduces to the Korteweg-de Vries equation, as well as the discrete integro-differential Painlevé II equation reduces to its continuous version. Our approach is based on the discrete analogue of Its-Izergin-Korepin-Slavnov theory of integrable operators developed by \textit{A. Borodin} [Int. Math. Res. Not. 2000, No. 9, 467--494 (2000; Zbl 0964.39015)]
and \textit{J. Baik} et al. [J. Am. Math. Soc. 12, No. 4, 1119--1178 (1999; Zbl 0932.05001)].Dynamics of stochastic FitzHugh-Nagumo system on unbounded domains with memoryhttps://zbmath.org/1528.370652024-03-13T18:33:02.981707Z"My, Bui Kim"https://zbmath.org/authors/?q=ai:my.bui-kim"Toan, Nguyen Duong"https://zbmath.org/authors/?q=ai:toan.nguyen-duongAuthors' abstract: In this paper, we consider the non-autonomous stochastic FitzHugh-Nagumo system with hereditary memory and a very large class of nonlinearities, which has no restriction on the upper growth of the nonlinearity. The existence of a random pullback attractor is established for this system in all \(N\)-dimensional space.
Reviewer: Anhui Gu (Chongqing)Limiting dynamics of stochastic heat equations with memory on thin domainshttps://zbmath.org/1528.370662024-03-13T18:33:02.981707Z"Shu, Ji"https://zbmath.org/authors/?q=ai:shu.ji"Li, Hui"https://zbmath.org/authors/?q=ai:li.hui.9"Huang, Xin"https://zbmath.org/authors/?q=ai:huang.xin.1"Zhang, Jian"https://zbmath.org/authors/?q=ai:zhang.jian(no abstract)Asymptotic stability of evolution systems of probability measures for nonautonomous stochastic systems: theoretical results and applicationshttps://zbmath.org/1528.370672024-03-13T18:33:02.981707Z"Wang, Renhai"https://zbmath.org/authors/?q=ai:wang.renhai"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan.Summary: The limiting stability of invariant probability measures of time homogeneous transition semigroups for autonomous stochastic systems has been extensively discussed in the literature. In this paper we initially initiate a program to study the asymptotic stability of evolution systems of probability measures of time inhomogeneous transition operators for nonautonomous stochastic systems. Two general theoretical results on this topic are established in a Polish space by establishing some sufficient conditions which can be verified in applications. Our abstract results are applied to a stochastic lattice reaction-diffusion equation driven by a time-dependent nonlinear noise. A time-average argument and an extended Krylov-Bogolyubov method due to \textit{G. da Prato} and \textit{M. Röckner} [Prog. Probab. 59, 115--122 (2008; Zbl 1154.60050)] are employed to prove the existence of evolution systems of probability measures. A mild condition on the time-dependent diffusion function is used to prove that the limit of every evolution system of probability measures must be an evolution system of probability measures of the limiting equation. The theoretical results are expected to be applied to various stochastic lattice systems/ODEs/PDEs in the future.Sharp growth of the Ornstein-Uhlenbeck operator on Gaussian tail spaceshttps://zbmath.org/1528.410252024-03-13T18:33:02.981707Z"Eskenazis, Alexandros"https://zbmath.org/authors/?q=ai:eskenazis.alexandros"Ivanisvili, Paata"https://zbmath.org/authors/?q=ai:ivanisvili.paataSummary: Let \(X\) be a standard Gaussian random variable. For any \(p \in (1, \infty)\), we prove the existence of a universal constant \(C_p> 0\) such that the inequality
\[
(\mathbb{E} | h^\prime (X) |^p)^{1/p} \geq C_p \sqrt{d} (\mathbb{E} | h(X)|^p)^{1/p}
\]
holds for all \(d \geq 1\) and all polynomials \(h : \mathbb{R} \to \mathbb{C}\) whose spectrum is supported on frequencies at least \(d\), that is, \(\mathbb{E}h (X) X^k = 0\) for all \(k = 0, 1, \dots, d-1\). As an application of this optimal estimate, we obtain an affirmative answer to the Gaussian analogue of a question of \textit{M. Mendel} and \textit{A. Naor} [Publ. Math., Inst. Hautes Étud. Sci. 119, 1--95 (2014; Zbl 1306.46021)] concerning the growth of the Ornstein-Uhlenbeck operator on tail spaces of the real line. We also show the same bound for the gradient of analytic polynomials in an arbitrary dimension.Rate of convergence in Trotter's approximation theorem and its applicationshttps://zbmath.org/1528.410382024-03-13T18:33:02.981707Z"Namba, Ryuya"https://zbmath.org/authors/?q=ai:namba.ryuyaSummary: The celebrated Trotter approximation theorem provides a sufficient condition for the convergence of a sequence of operator semigroups in terms of the corresponding sequence of infinitesimal generators. There exist a few results on the rate of convergence in Trotter's theorem under some constraints. In the present paper, a new rate of convergence in Trotter's theorem in full generality is given. Moreover, we see that this rate of convergence works well to obtain quantitative estimates for some limit theorems in probability theory.Log concavity preservation by beta operator based on probability toolshttps://zbmath.org/1528.410422024-03-13T18:33:02.981707Z"Badía, F. G."https://zbmath.org/authors/?q=ai:badia.francisco-german"Lee, H."https://zbmath.org/authors/?q=ai:lee.hyunju"Sangüesa, C."https://zbmath.org/authors/?q=ai:sanguesa.carmenSummary: A number of papers have dealt with the preservation of log convexity and log concavity based on various operators. For instance, in [\textit{F. G. Badía} and \textit{C. Sangüesa}, J. Math. Anal. Appl. 413, No. 2, 953--962 (2014; Zbl 1310.41012)], the preservation of log convexity and log concavity under Bernstein operators was discussed based on some characteristics of a stochastic process. However, regarding beta-type operators, the preservation of log concavity does not hold based on such a probabilistic method used in [loc. cit.]. In this study, we explore the preservation of log concavity for the beta operator using alternative probabilistic tools. Notably, we show results on the preservation of log concavity for monotone log concave functions. Further, some results of application to some specific functions, ageing classes of the deterioration Dirichlet process, related operators and order statistics are provided.Polynomial approximations in a generalized Nyman-Beurling criterionhttps://zbmath.org/1528.410472024-03-13T18:33:02.981707Z"Alouges, François"https://zbmath.org/authors/?q=ai:alouges.francois"Darses, Sébastien"https://zbmath.org/authors/?q=ai:darses.sebastien"Hillion, Erwan"https://zbmath.org/authors/?q=ai:hillion.erwanSummary: The Nyman-Beurling criterion, equivalent to the Riemann hypothesis (RH), is an approximation problem in the space of square integrable functions on \((0, \infty)\), involving dilations of the fractional part function by factors \(\theta_k\in(0, 1)\), \(k \geq 1\). Randomizing the \(\theta_k\) gener
By the way, I had editing rights over that page, maybe they should be
transferred to you. The person I was in touch with was '''Robert
Gieseke''' <>ates new structures and criteria. One of them is a sufficient condition for RH that splits into
\begin{itemize}
\item[(i)] showing that the indicator function can be approximated by convolution with the fractional part,
\item[(ii)] a control on the coefficients of the approximation.
\end{itemize}
This self-contained paper generalizes conditions (i) and (ii) that involve a \(\sigma_0\in(1/2, 1)\), and imply \(\zeta (\sigma +it) \neq 0\) in the strip \(1/2 < \sigma \leq \sigma_0 < 1\). We then identify functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In this context, the difficulty for proving RH is then reallocated in (ii), which heavily relies on the corresponding Gram matrices, for which two remarkable structures are obtained. We show that a particular tuning of the approximating sequence leads to a striking simplification of the second Gram matrix, then reading as a block Hankel form.Exponential moments for disk counting statistics of random normal matrices in the critical regimehttps://zbmath.org/1528.410812024-03-13T18:33:02.981707Z"Charlier, Christophe"https://zbmath.org/authors/?q=ai:charlier.christophe"Lenells, Jonatan"https://zbmath.org/authors/?q=ai:lenells.jonatanSummary: We obtain large \(n\) asymptotics for the \(m\)-point moment generating function of the disk counting statistics of the Mittag-Leffler ensemble, where \(n\) is the number of points of the process and \(m\) is arbitrary but fixed. We focus on the critical regime where all disk boundaries are merging at speed \(n^{-\frac{1}{2}}\), either in the bulk or at the edge. As corollaries, we obtain two central limit theorems and precise large \(n\) asymptotics of all joint cumulants (such as the covariance) of the disk counting function. Our results can also be seen as large \(n\) asymptotics for \(n\times n\) determinants with merging planar discontinuities.On the best constant in the \(L^p\) estimate for the sharp maximal functionhttps://zbmath.org/1528.420242024-03-13T18:33:02.981707Z"Kamiński, Łukasz"https://zbmath.org/authors/?q=ai:kaminski.lukasz"Osękowski, Adam"https://zbmath.org/authors/?q=ai:osekowski.adamThis article studies the best constant in the \(L^p(\mathbb{R})\) estimate for the sharp maximal function \(\widetilde{f^\#}\) associated with the bounded lower oscillation setting. This maximal function is given by the formula \(\widetilde{f^\#}(x)=\underset{I}{\sup}\big(\langle f\rangle-\underset{I}{\mbox{ ess inf}}(f)\big)\) where \(I\) is an interval of \(\mathbb{R}\) and the \(L^p\) estimate studied here is \(\|f\|_{L^p}\leq C_p\|\widetilde{f^\#}\|_{L^p}\). The main theorem of this article states that if \(f\in L^p(I)\) for \(1\leq p<+\infty\) where \(I\) is an arbitrary interval and if \(f\) is of integral zero, then the best constant \(C_p\) in the previous estimate is \(p-1\) if \(p\geq 2\) and \(\displaystyle{\left(\int_0^{+\infty}e^{-\rho}|\rho-1|^pd\rho\right)^\frac{1}{p}}\) if \(1\leq p<2\). The proof of the theorem uses explicit evaluations of the associated Bellman function.
Reviewer: Diego Chamorro (Évry)Ergodicity of exclusion semigroups constructed from quantum Bernoulli noiseshttps://zbmath.org/1528.460512024-03-13T18:33:02.981707Z"Chen, Jinshu"https://zbmath.org/authors/?q=ai:chen.jinshu"Hai, Shexiang"https://zbmath.org/authors/?q=ai:hai.shexiangSummary: Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. This paper aimed to discuss the classical reduction and ergodicity of quantum exclusion semigroups constructed by QBN. We first study the classical reduction of the quantum semigroups to an abelian algebra of diagonal elements and the space of off-diagonal elements. We then provide an explicit representation formula by separating the action on off-diagonal and diagonal operators, on which they are reduced to the semigroups of classical Markov chains. Finally, we prove that the asymptotic behavior of the quantum semigroups is equivalent to one of its associated Markov chains, and that the semigroups restricted to the off diagonal space of operators have a zero limit.Decomposable product systems associated to non-stationary Poisson processeshttps://zbmath.org/1528.460572024-03-13T18:33:02.981707Z"Shanmugasundaram, Sundar"https://zbmath.org/authors/?q=ai:shanmugasundaram.sundarSummary: Let \(P\) be a closed convex cone in \(\mathbb{R}^d\), which we assume is pointed and spanning, i.e., \(P\cap(-P)=\{0\}\) and \(P-P=\mathbb{R}^d\). We demonstrate that, when \(d\geq 2\), in contrast to the one-parameter situation, Poisson processes on \(\mathbb{R}^d\), with intensity measure absolutely continuous with respect to the Lebesgue measure, restricted to \(P\)-invariant closed subsets, provide us with a source of examples of decomposable \(E_0\)-semigroups that are not always CCR flows.Stochastic applications of Caputo-type convolution operators with nonsingular kernelshttps://zbmath.org/1528.470032024-03-13T18:33:02.981707Z"Beghin, Luisa"https://zbmath.org/authors/?q=ai:beghin.luisa"Caputo, Michele"https://zbmath.org/authors/?q=ai:caputo.micheleSummary: We consider here convolution operators, in the Caputo sense, with nonsingular kernels. We prove that the solutions to some integro-differential equations with such operators (acting on the space variable) coincide with the transition densities of a particular class of Lévy subordinators (i.e. compound Poisson processes with non-negative jumps). We then extend these results to the case where the kernels of the operators have random parameters, with given distribution. This assumption allows greater flexibility in the choice of the kernel's parameters and, consequently, of the jumps' density function.Approximate optimal control of fractional impulsive partial stochastic differential inclusions driven by Rosenblatt processhttps://zbmath.org/1528.490232024-03-13T18:33:02.981707Z"Yan, Zuomao"https://zbmath.org/authors/?q=ai:yan.zuomaoSummary: In this paper, we study the approximate optimal control problems for a class of fractional partial stochastic differential inclusions driven by Rosenblatt process and non-instantaneous impulses in a Hilbert space. Firstly, we prove an existence result of mild solutions for the control systems by using stochastic analysis, the fractional calculus, the measure of noncompactness, properties of sectorial operators and fixed point theorems. Secondly, we derive the existence conditions of approximate solutions to optimal control problems governed by fractional impulsive partial stochastic differential control systems with the help of minimizing sequence method. Finally, an example is given for the illustration of the obtained theoretical results.Large deviation principle and thermodynamic limit of chemical master equation via nonlinear semigrouphttps://zbmath.org/1528.490262024-03-13T18:33:02.981707Z"Gao, Yuan"https://zbmath.org/authors/?q=ai:gao.yuan|gao.yuan.1"Liu, Jian-Guo"https://zbmath.org/authors/?q=ai:liu.jian-guoSummary: Chemical reactions can be modeled by a random time-changed Poisson process on countable states. The macroscopic behaviors, such as large fluctuations, can be studied via the WKB reformulation. The WKB reformulation for the backward equation is Varadhan's discrete nonlinear semigroup and is also a monotone scheme that approximates the limiting first-order Hamilton-Jacobi equations (HJE). The discrete Hamiltonian is an \(m\)-accretive operator, which generates a nonlinear semigroup on countable grids and justifies the well-posedness of the chemical master equation and the backward equation with ``no reaction'' boundary conditions. The convergence from the monotone schemes to the viscosity solution of HJE is proved by constructing barriers to overcome the polynomial growth coefficients in the Hamiltonian. This implies the convergence of Varadhan's discrete nonlinear semigroup to the continuous Lax-Oleinik semigroup and leads to the large deviation principle for the chemical reaction process at any single time. Consequently, the macroscopic mean-field limit reaction rate equation is recovered with a concentration rate estimate. Furthermore, we establish the convergence from a reversible invariant measure to an upper semicontinuous viscosity solution of the stationary HJE.The volume of random simplices from elliptical distributions in high dimensionhttps://zbmath.org/1528.520022024-03-13T18:33:02.981707Z"Gusakova, Anna"https://zbmath.org/authors/?q=ai:gusakova.anna"Heiny, Johannes"https://zbmath.org/authors/?q=ai:heiny.johannes"Thäle, Christoph"https://zbmath.org/authors/?q=ai:thale.christophThe aim of the present paper is to provide central limit theorems for the logarithmic volume of \(p\)-dimensional pinned random simplices whose generating points follow a general elliptical distribution in \(\mathbb{R}^n\). In particular, the authors are interested in the high-dimensional regime, where \(p\) is a function of the space dimension \(n\) satisfying \(p\to +\infty\), as \(n\to +\infty\), and is such that \(p/n \to \gamma \in (0, 1)\). An \(n\)-dimensional random vector \(\mathbf{x}\) follows an elliptical distribution if it takes the form \(\mathbf{x} = R\mathbf{A}\mathbf{u}\), where \(R \ge 0\) is an arbitrary real-valued random variable, \(\mathbf{A}\) is a fixed \(n\times n\) matrix of full rank and \(\mathbf{u}\) is a uniform random direction, that is, a uniform random point on the unit sphere in \(\mathbb{R}^n\), which is independent from \(R\). A variety of different models and results are known in the literature, mainly in the asymptotic regime \(p \to +\infty\), while the dimension parameter \(n\) is kept fixed. Examples include the expectation asymptotics for the number of faces or the intrinsic volumes and their tight relation to affine surface areas, related upper and lower variance bounds as well as results on the asymptotic normality or concentration properties for these combinatorial and geometric parameters, see [\textit{D. Hug}, Lect. Notes Math. 2068, 205--238 (2013; Zbl 1275.60017)] for motivation, background material and references.
The paper is organized as follows. Section 2 introduces the authors' set-up together with the necessary notation. The mail results, Theorem 2.1 for the log-determinant (asymptotic normality in high dimensions) and its geometric counterpart Theorem 2.4, are the contents of Section 2.1 and Section 2.2, respectively. Section 3 is devoted to the proof of Theorem 2.1, while the Appendix contains some auxiliary lemmas.
Reviewer: Viktor Ohanyan (Erevan)Projections and angle sums of belt polytopes and permutohedrahttps://zbmath.org/1528.520052024-03-13T18:33:02.981707Z"Godland, Thomas"https://zbmath.org/authors/?q=ai:godland.thomas"Kabluchko, Zakhar"https://zbmath.org/authors/?q=ai:kabluchko.zakhar-aA polytope \(P\subset {\mathbb{R}}^n\) is a \textit{belt polytope} if its normal fan coincides with the fan of some hyperplane arrangement \(\mathcal{A}\). An affine subspace \(M\subset {\mathbb{R}}^n\) is in \textit{general position w.r.t.~a linear subspace \(L\subset {\mathbb{R}}^n\)} if the unique linear space \(M'\) through origin and parallel to \(M\) satisfies \(\dim(M'\cap L) = \max(\dim(M') + \dim(L) - n,0)\). A linear subspace \(L\subset {\mathbb{R}}^n\) is in \textit{general position w.r.t.~polyhedral set \(P\)} if the affine hull of each face \(F\) of \(P\) is in general positing w.r.t.~\(L\).
Let \(P\subset {\mathbb{R}}^n\) be a belt polytope and \(G : {\mathbb{R}}^n \rightarrow {\mathbb{R}}^d\) be a surjective linear map, so \(\dim(\ker(G))=n-d\). If \(\ker(G)\subset {\mathbb{R}}^n\) is in general position w.r.t.~the faces of \(P\), then a formula for the number of \(j\)-dimensional faces of the projected polytope \(GP\) is derived in terms of the \(j\)-level characteristic polynomial of the hyperplane arrangement \(\mathcal{A}\). Here, the \textit{characteristic polynomial} of \(\mathcal{A}\) is given by
\[
\chi_{\mathcal{A}}(t) = \sum_{\mathcal{C}\subset\mathcal{A}} (-1)^{|\mathcal{C}|}\ t^{\dim(I_{\mathcal{C}})},
\]
where \(I_{\mathcal{C}} = \bigcap_{H\in\mathcal{C}}H\) is the intersection of all the hyperplanes in \(\mathcal{C}\), and the \textit{\(j\)-th level characteristic polynomial} of \(\mathcal{A}\) is defined by
\[
\chi_{{\mathcal{A}},j}(t) := \sum_{M\in {\mathcal{L}}_j(\mathcal{A})}\chi_{\mathcal{A}|M}(t),
\]
where \({\mathcal{L}}_j(\mathcal{A})\) is the sublattice consisting of all the \(j\)-dimensional linear subspaces from the lattice \(\mathcal{L}(\mathcal{A})\) of all linear subspaces of \(\mathcal{A}\) and their intersections, and \({\mathcal{A}|M} := \{H\cap M : H\in \mathcal{A}, \ \ M\not\subseteq H\}\). In particular, it is shown that the face numbers of \(GP\) do not depend on on the linear map \(G\) provided that \(G\) satisfies some general and natural assumptions.
For a polyhedral set \(P\subset {\mathbb{R}}\) the \textit{tangent cone} \(T_f(P)\) of a face \(F\) of \(P\) is defined by \(T_F(P) = \{\tilde{x}\in {\mathbb{R}}^n : \tilde{x}_0 + \epsilon\tilde{x}\in P \mbox{ for some }\epsilon >0\}\) where \(\tilde{x}_0\) is some point in the relative interior of the face \(F\). The paper further derives formulas for the sum of the conic intrinsic volumes and Grassmann angles of the tangent cones of \(P\) at all of its \(j\)-faces. Here, the \textit{conic intrinsic volumes} are the analogues of the usual intrinsic volumes in the setting of conical or spherical geometry and can be defined using the spherical Steiner formula, and the \textit{Grassmann angles} \(\gamma_k(C)\) of a given cone \(C\subset {\mathbb{R}}^n\) are defined in terms of a random linear subspace of dimension \(n-k\) with uniform distributions on the Grassmannian of all \((n-k)\)-dimensional subspaces of \({\mathbb{R}}^n\). These mentioned formulas are then applied to permutohedra of type \(A\) and \(B\) which then yield closed formulas for the face numbers of the projected permutohedra and the generalized angle sums of permutohedra in terms of Stirling numbers of both first and the second kind. Here, a permutohedron of type \(A\) in terms of \(\tilde{x} = (x_1,\dots,x_n)\in {\mathbb{R}}^n\) is defined by \({\mathcal{P}}_n^A(\tilde{x}) := \mathrm{conv}\{(x_{\sigma(1)},\dots,x_{\sigma(n)}) : \sigma\in S_n\}\) and a permutohedron of type \(B\) in terms of \(\tilde{x} = (x_1,\dots,x_n)\in {\mathbb{R}}^n\) is defined by \({\mathcal{P}}_n^B(\tilde{x}) := \mathrm{conv}\{(\epsilon_1x_{\sigma(1)},\dots,\epsilon_nx_{\sigma(n)}) : (\epsilon_1,\dots,\epsilon_n)\in \{\pm 1\}^n, \ \ \sigma\in S_n\}\), where \(S_n\) is the symmetric group of all permutations on \(n\) letters. These permutohedra of type \(A\) and \(B\) are examples of belt polytopes.
Reviewer: Geir Agnarsson (Fairfax)Geometric science of information. 6th international conference, GSI 2023, St. Malo, France, August 30 -- September 1, 2023. Proceedings. Part IIhttps://zbmath.org/1528.530022024-03-13T18:33:02.981707ZThe articles of this volume will be reviewed individually. For the preceding conference see [Zbl 1482.94007]. For Part II of the proceedings of the present conference see [Zbl 1528.94003].
Indexed articles:
\textit{Terze, Zdravko; Zlatar, Dario; Kasalo, Marko; Andrić, Marijan}, Lie group quaternion attitude-reconstruction of quadrotor UAV, 3-11 [Zbl 07789278]
\textit{de Saxcé, Géry}, A variational principle of minimum for Navier-Stokes equation based on the symplectic formalism, 12-21 [Zbl 07789279]
\textit{Dubois, François; Rojas-Quintero, Juan Antonio}, A variational symplectic scheme based on Simpson's quadrature, 22-31 [Zbl 07789280]
\textit{Bergshoeff, Eric}, Generalized Galilean geometries, 32-40 [Zbl 07789281]
\textit{Crespo, Mewen; Casale, Guy; Le Marrec, Loïc}, Continuum mechanics of defective media: an approach using fiber bundles, 41-49 [Zbl 07789282]
\textit{Coquinot, Baptiste; Mir, Pau; Miranda, Eva}, Singular cotangent models in fluids with dissipation, 50-59 [Zbl 07789283]
\textit{Li, Tianzhi; Wang, Jinzhi}, Multisymplectic unscented Kalman filter for geometrically exact beams, 60-68 [Zbl 07789284]
\textit{Cardall, Christian Y.}, Towards full `Galilei general relativity': Bargmann-Minkowski and Bargmann-Galilei spacetimes, 69-78 [Zbl 07789285]
\textit{Colombo, Leonardo; Goodman, Jacob}, Existence of global minimizer for elastic variational obstacle avoidance problems on Riemannian manifolds, 81-88 [Zbl 07789286]
\textit{Stratoglou, Efstratios; Anahory Simoes, Alexandre; Bloch, Anthony; Colombo, Leonardo}, Virtual affine nonholonomic constraints, 89-96 [Zbl 07789287]
\textit{Simoes, Alexandre Anahory; Colombo, Leonardo}, Nonholonomic systems with inequality constraints, 97-104 [Zbl 07789288]
\textit{de León, Manuel; Lainz, Manuel; López-Gordón, Asier; Marrero, Juan Carlos}, Nonholonomic brackets: Eden revisited, 105-112 [Zbl 07789289]
\textit{Cushman, Richard}, The momentum mapping of the affine real symplectic group, 115-123 [Zbl 07789290]
\textit{El Morsalani, Mohamed}, Polysymplectic Souriau Lie group thermodynamics and the geometric structure of its coadjoint orbits, 124-133 [Zbl 07789291]
\textit{El Morsalani, Mohamed}, Polysymplectic Souriau Lie group thermodynamics and entropy geometric structure as Casimir invariant function, 134-143 [Zbl 07789292]
\textit{Bieliavsky, Pierre; Dendoncker, Valentin; Neuttiens, Guillaume; Pierard de Maujouy, Jérémie}, Riemannian geometry of Gibbs cones associated to nilpotent orbits of simple Lie groups, 144-151 [Zbl 07789293]
\textit{Barbaresco, Frédéric}, Symplectic foliation transverse structure and Libermann foliation of heat theory and information geometry, 152-164 [Zbl 07789294]
\textit{Combe, Noemie; Combe, Philippe; Nencka, Hanna}, Poisson geometry of the statistical Frobenius manifold, 165-172 [Zbl 07789295]
\textit{Boyom, Michel Nguiffo}, Canonical Hamiltonian systems in symplectic statistical manifolds, 173-180 [Zbl 07789296]
\textit{Naudts, Jan}, A dually flat geometry for spacetime in 5d, 183-191 [Zbl 07789297]
\textit{Bendimerad-Hohl, Antoine; Haine, Ghislain; Matignon, Denis}, Structure-preserving discretization of the Cahn-Hilliard equations recast as a port-Hamiltonian system, 192-201 [Zbl 07789298]
\textit{Rodríguez Abella, Álvaro; Gay-Balmaz, François; Yoshimura, Hiroaki}, Infinite dimensional Lagrange-Dirac mechanics with boundary conditions, 202-211 [Zbl 07789299]
\textit{Wu, Meng; Gay-Balmaz, François}, Variational integrators for stochastic Hamiltonian systems on Lie groups, 212-220 [Zbl 07789300]
\textit{Yoshimura, Hiroaki; Gay-Balmaz, François}, Hamiltonian variational formulation for non-simple thermodynamic systems, 221-230 [Zbl 07789301]
\textit{Bergold, Paul; Tronci, Cesare}, Madelung transform and variational asymptotics in Born-Oppenheimer molecular dynamics, 231-241 [Zbl 07789302]
\textit{Castrillón López, Marco}, Conservation laws as part of Lagrangian reduction. Application to image evolution, 242-250 [Zbl 07789303]
\textit{Bauer, Werner; Brecht, Rüdiger}, Casimir-dissipation stabilized stochastic rotating shallow water equations on the sphere, 253-262 [Zbl 07789304]
\textit{Possanner, Stefan; Holderied, Florian; Li, Yingzhe; Na, Byung Kyu; Bell, Dominik; Hadjout, Said; Güçlü, Yaman}, High-order structure-preserving algorithms for plasma hybrid models, 263-271 [Zbl 07789305]
\textit{Spera, Mauro}, Hydrodynamics of the probability current in Schrödinger theory, 272-281 [Zbl 07789306]
\textit{Gay-Balmaz, François; Wu, Meng; Eldred, Chris}, Variational geometric description for fluids with permeable boundaries, 282-289 [Zbl 07789307]
\textit{Tronci, Cesare; Gay-Balmaz, François}, Lagrangian trajectories and closure models in mixed quantum-classical dynamics, 290-300 [Zbl 07789308]
\textit{Vizman, Cornelia}, A discrete version for vortex loops in 2D fluids, 301-309 [Zbl 07789309]
\textit{Ortega, Juan-Pablo; Yin, Daiying}, Expressiveness and structure preservation in learning port-Hamiltonian systems, 313-322 [Zbl 07789310]
\textit{Chrétien, Stéphane; Vaucher, Rémi}, Signature estimation and signal recovery using median of means, 323-331 [Zbl 07789311]
\textit{Toshev, Artur P.; Galletti, Gianluca; Brandstetter, Johannes; Adami, Stefan; Adams, Nikolaus A.}, Learning Lagrangian fluid mechanics with E(3)-equivariant graph neural networks, 332-341 [Zbl 07789312]
\textit{Jiang, Haotian; Li, Qianxiao}, Forward and inverse approximation theory for linear temporal convolutional networks, 342-350 [Zbl 07789313]
\textit{Nakamura, Takemi}, Monotonicity of the scalar curvature of the quantum exponential family for transverse-field Ising chains, 353-362 [Zbl 07789314]
\textit{Ciaglia, F. M.; Di Cosmo, F.; González-Bravo, L.}, Can Čencov meet Petz, 363-371 [Zbl 07789315]
\textit{Barbaresco, Frederic}, Souriau's geometric principles for quantum mechanics, 372-381 [Zbl 07789316]
\textit{Jacquet, Philippe; Joly, Véronique}, Unitarity excess in Schwartzschild metric, 382-391 [Zbl 07789317]
\textit{Verrier, Gabriel; Haine, Ghislain; Matignon, Denis}, Modelling and structure-preserving discretization of the Schrödinger as a port-Hamiltonian system, and simulation of a controlled quantum box, 392-401 [Zbl 07789318]
\textit{Ciaglia, Florio M.; Di Cosmo, Fabio}, Some remarks on the notion of transition, 402-411 [Zbl 07789319]
\textit{de Gosson, Maurice A.}, Geometric quantum states and Lagrangian polar duality: quantum mechanics without wavefunctions, 412-419 [Zbl 07789320]
\textit{Mama Assandje, Prosper Rosaire; Dongho, Joseph; Bouetou, Thomas Bouetou}, Complete integrability of gradient systems on a manifold admitting a potential in odd dimension, 423-432 [Zbl 07789321]
\textit{Drumetz, Lucas; Reiffers-Masson, Alexandre; El Bekri, Naoufal; Vermet, Franck}, Geometry-preserving Lie group integrators for differential equations on the manifold of symmetric positive definite matrices, 433-443 [Zbl 07789322]
\textit{Uwano, Yoshio}, The phase space description of the geodesics on the statistical model on a finite set. Trajectory-confinement and integrability, 444-453 [Zbl 07789323]
\textit{Tarama, Daisuke; Françoise, Jean-Pierre}, Geodesic flows of \(\alpha \)-connections for statistical transformation models on a compact Lie group, 454-462 [Zbl 07789324]
\textit{Galeotti, Mattia; Citti, Giovanna; Sarti, Alessandro}, Differential operators heterogenous in orientation and scale in the \(V_1\) cortex, 465-473 [Zbl 07789325]
\textit{Liontou, Vasiliki}, Gabor frames and contact structures: signal encoding and decoding in the primary visual cortex, 474-482 [Zbl 07789326]
\textit{Mazzetti, Caterina; Sarti, Alessandro; Citti, Giovanna}, A sub-Riemannian model of the functional architecture of M1 for arm movement direction, 483-492 [Zbl 07789327]
\textit{Alekseevsky, D. V.; Shirokov, I. M.}, Geometry of saccades and saccadic cycles, 493-500 [Zbl 07789328]
\textit{Tamekue, Cyprien; Prandi, Dario; Chitour, Yacine}, MacKay-type visual illusions via neural fields, 501-508 [Zbl 07789329]
\textit{Da Rocha, Wagner; Lavor, Carlile; Liberti, Leo; Malliavin, Thérèse E.}, Pseudo-dihedral angles in proteins providing a new description of the Ramachandran map, 511-519 [Zbl 07789330]
\textit{Hengeveld, Simon B.; Merabti, Mathieu; Pascale, Fabien; Malliavin, Thérèse E.}, A study on the covalent geometry of proteins and its impact on distance geometry, 520-530 [Zbl 07789331]
\textit{Huang, Shu-Yu; Chang, Chi-Fon; Lin, Jung-Hsin; Malliavin, Thérèse E.}, Exploration of conformations for an intrinsically disordered protein, 531-540 [Zbl 07789332]
\textit{Tumpach, Alice Barbora; Kán, Peter}, Temporal alignment of human motion data: a geometric point of view, 541-550 [Zbl 07789333]
\textit{Hengeveld, S. B.; Plastria, F.; Mucherino, A.; Pelta, D. A.}, A linear program for points of interest relocation in adaptive maps, 551-559 [Zbl 07789334]
\textit{Wohrer, Adrien}, Diffeomorphic ICP registration for single and multiple point sets, 563-573 [Zbl 07789335]
\textit{Makaroff, Nicolas; Cohen, Laurent D.}, Chan-Vese attention U-net: an attention mechanism for robust segmentation, 574-582 [Zbl 07789336]
\textit{Li, Wanxin; Prasad, Ashok; Miolane, Nina; Dao Duc, Khanh}, Using a Riemannian elastic metric for statistical analysis of tumor cell shape heterogeneity, 583-592 [Zbl 07789337]
\textit{Lefevre, Julien; Fraize, Justine; Germanaud, David}, Perturbation of Fiedler vector: interest for graph measures and shape analysis, 593-601 [Zbl 07789338]
\textit{Kratsios, Anastasis; Hyndman, Cody}, Generative OrnsteinUhlenbeck markets via geometric deep learning, 605-614 [Zbl 07789339]
\textit{Lagrave, Pierre-Yves}, \(\mathrm{SL}(2,\mathbb{Z})\)-equivariant machine learning with modular forms theory and applications, 615-623 [Zbl 07789340]
\textit{Camarinha, Margarida; Machado, Luís; Silva Leite, Fátima}, K-splines on SPD manifolds, 624-633 [Zbl 07789341]
\textit{Lapenna, M.; Faglioni, F.; Zanchetta, F.; Fioresi, R.}, Geometric deep learning: a temperature based analysis of graph neural networks, 634-643 [Zbl 07789342]Fold maps associated to geodesic random walks on non-positively curved manifoldshttps://zbmath.org/1528.570232024-03-13T18:33:02.981707Z"Lessa, Pablo"https://zbmath.org/authors/?q=ai:lessa.pablo"Oliveira, Lucas"https://zbmath.org/authors/?q=ai:oliveira.lucas-sSummary: We study a family of mappings from the powers of the unit tangent sphere at a point to a complete Riemannian manifold with non-positive sectional curvature, whose behavior is related to the spherical mean operator and the geodesic random walks on the manifold.
We show that for odd powers of the unit tangent sphere the mappings are fold maps.
Some consequences on the regularity of the transition density of geodesic random walks, and on the eigenfunctions of the spherical mean operator are discussed and related to previous work.Malliavin calculus for Lévy processes with applications to financehttps://zbmath.org/1528.600012024-03-13T18:33:02.981707Z"Di Nunno, Giulia"https://zbmath.org/authors/?q=ai:di-nunno.giulia"Øksendal, Bernt"https://zbmath.org/authors/?q=ai:oksendal.bernt-karsten"Proske, Frank"https://zbmath.org/authors/?q=ai:proske.frank-norbertPublisher's description: While the original works on Malliavin calculus aimed to study the smoothness of densities of solutions to stochastic differential equations, this book has another goal. It portrays the most important and innovative applications in stochastic control and finance, such as hedging in complete and incomplete markets, optimisation in the presence of asymmetric information and also pricing and sensitivity analysis. In a self-contained fashion, both the Malliavin calculus with respect to Brownian motion and general Lévy type of noise are treated.
Besides, forward integration is included and indeed extended to general Lévy processes. The forward integration is a recent development within anticipative stochastic calculus that, together with the Malliavin calculus, provides new methods for the study of insider trading problems.
To allow more flexibility in the treatment of the mathematical tools, the generalization of Malliavin calculus to the white noise framework is also discussed.
This book is a valuable resource for graduate students, lecturers in stochastic analysis and applied researchers.Mean-square approximation of iterated Ito and Stratonovich stochastic integrals: method of generalized multiple Fourier series. Application to numerical integration of Ito SDEs and semilinear SPDEshttps://zbmath.org/1528.600022024-03-13T18:33:02.981707Z"Kuznetsov, Dmitriy Feliksovich"https://zbmath.org/authors/?q=ai:kuznetsov.dmitriy-feliksovichSummary: This is the third edition of the monograph (first edition [the author, Differ. Uravn. Protsessy Upr. 2020, No. 4, 606~p. (2020; Zbl 1456.65001)] , second edition [the author, Differ. Uravn. Protsessy Upr. 2021, No. 4, 788~p. (2021; Zbl 1492.60004)]) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the first time we use the generalized multiple Fourier series converging in the sense of norm in Hilbert space for the expansion of iterated Ito stochastic integrals of arbitrary multiplicity \(k\) with respect to components of the multidimensional Wiener process (Chapter 1). Sections 1.11--1.13 (Chapter 1) are new and generalize the results of Chapter 1 obtained earlier by the author and are also closely related to the multiple Wiener stochastic integral introduced by Ito in 1951. The convergence with probability 1 as well as the convergence in the sense of \(n\)-th (\(n=2, 3,\dots\)) moment for the expansion of iterated Ito stochastic integrals have been proved (Chapter 1). Moreover, the rate of both types of convergence has been established. The main difference between the third and second editions of the book is that the third edition includes original material (Chapter 2, Sections 2.10--2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity \(k\) with respect to components of the multidimensional Wiener process. The above approach allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. In addition, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. These results (Chapter 2) adapt the results of Chapter 1 for iterated Stratonovich stochastic integrals. Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity \(k\) based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity \(k\) (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). These approximations were obtained using Legendre polynomials and trigonometric functions. The methods constructed in the book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity \(k\)) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).
For the first edition see [Zbl 1492.60004].A conversation with Paul Embrechtshttps://zbmath.org/1528.600032024-03-13T18:33:02.981707Z"Genest, Christian"https://zbmath.org/authors/?q=ai:genest.christian"Nešlehová, Johanna G."https://zbmath.org/authors/?q=ai:neslehova.johanna-gSummary: Paul Embrechts was born in Schoten, Belgium, on 3 February 1953. He holds a Licentiaat in Mathematics from Universiteit Antwerpen (1975) and a DSc from Katholieke Universiteit Leuven (1979), where he was also a Research Assistant from 1975 to 1983. He then held a lectureship in Statistics at Imperial College, London (1983-1985) and was a Docent at Limburgs Universitair Centrum, Belgium (1985-1989) before joining ETH Zürich as a Full Professor of Mathematics in 1989, where he remained until his retirement as an Emeritus in 2018. A renowned specialist of extreme-value theory and quantitative risk management, he authored or coauthored nearly 200 scientific papers and five books, including the highly influential `Modelling of Extremal Events for Insurance and Finance' (Springer, 1997) and `Quantitative Risk Management: Concepts, Techniques and Tools' (Princeton University Press, 2005, 2015). He served in numerous editorial capacities, notably as Editor-in-Chief of the (1996-2005). Praised for his natural leadership and exceptional communication skills, he helped to bridge the gap between academia and industry through the foundation of RiskLab Switzerland and his sustained leadership for nearly 20 years. He gave numerous prestigious invited and keynote lectures worldwide and served as a member of the board of, or consultant for, various banks, insurance companies and international regulatory authorities. His work was recognised through several visiting positions, including at the Oxford-Man Institute, and many awards. He is, inter alia, an Elected Fellow of the Institute of Mathematical Statistics (1995) and the American Statistical Association (2014), an Honorary Fellow of the Institute and the Faculty of Actuaries (2000), Honorary Member of the Belgian (2010) and French (2015) Institute of Actuaries and was granted four honorary degrees (University of Waterloo, 2007; Heriot-Watt University, 2011; Université catholique de Louvain, 2012; City, University of London, 2017). The following conversation took place in Paul's office at ETH Zürich, 17-18 December 2018.
{{\copyright} 2020 The Authors. International Statistical Review published by John Wiley \& Sons Ltd on behalf of International Statistical Institute.}The future of probabilityhttps://zbmath.org/1528.600042024-03-13T18:33:02.981707Z"Protter, Philip"https://zbmath.org/authors/?q=ai:protter.philip-eSummary: Probability as a subject in and of itself has rarely been truly appreciated by mathematicians in other disciplines. This has gradually changed over the last 50 years, as occasionally brilliant mathematicians show how it can be used to solve, or to explain, and/or to give intuitive content to thorny mathematical issues. We provide some examples and then give a wild speculation as to where the field, at least in Mathematical Finance, might go in the future.
For the entire collection see [Zbl 1515.01005].Abstracts of talks given at the 7th international conference on stochastic methods. IIhttps://zbmath.org/1528.600052024-03-13T18:33:02.981707Z"Shiryaev, A. N."https://zbmath.org/authors/?q=ai:shiryaev.albert-n"Pavlov, I. V."https://zbmath.org/authors/?q=ai:pavlov.igor-vSummary: This is the second installment of a two-part article presenting abstracts of talks given at the 7th International Conference on Stochastic Methods (ICSM-7), held June 2-9, 2022 at Divnomorskoe (near the town of Gelendzhik) at the Raduga sports and fitness center of the Don State Technical University. The conference was chaired by A. N. Shiryaev. Participants included leading scientists from Russia, France, Portugal, and Tadjikistan.On the pragmatic and epistemic virtues of inference to the best explanationhttps://zbmath.org/1528.600062024-03-13T18:33:02.981707Z"Pettigrew, Richard"https://zbmath.org/authors/?q=ai:pettigrew.richardSummary: In a series of papers over the past twenty years, and in a new book, Igor Douven (sometimes in collaboration with Sylvia Wenmackers) has argued that Bayesians are too quick to reject versions of inference to the best explanation that cannot be accommodated within their framework. In this paper, I survey their worries and attempt to answer them using a series of pragmatic and purely epistemic arguments that I take to show that Bayes' Rule really is the only rational way to respond to your evidence.A classical way forward for the regularity and normalization problemshttps://zbmath.org/1528.600072024-03-13T18:33:02.981707Z"Pruss, Alexander R."https://zbmath.org/authors/?q=ai:pruss.alexander-rSummary: Bayesian epistemology has struggled with the problem of regularity: how to deal with events that in classical probability have zero probability. While the cases most discussed in the literature, such as infinite sequences of coin tosses or continuous spinners, do not actually come up in scientific practice, there are cases that do come up in science. I shall argue that these cases can be resolved without leaving the realm of classical probability, by choosing a probability measure that preserves ``enough'' regularity. This approach also provides a resolution to the McGrew, McGrew and Vestrum normalization problem for the fine-tuning argument.Bernoulli's golden theorem in retrospect: error probabilities and trustworthy evidencehttps://zbmath.org/1528.600082024-03-13T18:33:02.981707Z"Spanos, Aris"https://zbmath.org/authors/?q=ai:spanos.arisSummary: Bernoulli's 1713 golden theorem is viewed retrospectively in the context of modern model-based frequentist inference that revolves around the concept of a prespecified statistical model \(\mathcal{M}_{\theta}(\mathbf{x})\), defining the inductive premises of inference. It is argued that several widely-accepted claims relating to the golden theorem and frequentist inference are either misleading or erroneous: (a) Bernoulli solved the problem of inference `from probability to frequency', and thus (b) the golden theorem cannot justify an approximate Confidence Interval (CI) for the unknown parameter \(\theta \), (c) Bernoulli identified the probability \(P(A)\) with the relative frequency \(\frac{1}{n}\sum\nolimits_{k = 1}^n {x_k }\) of event \(A\) as a result of conflating \(f(\mathbf{x}_0 |\theta)\) with \(f(\theta |\mathbf{x}_0)\), where \(\mathbf{x}_0\) denotes the observed data, and (d) the same `swindle' is currently perpetrated by the \(p\) value testers. In interrogating the claims (a)--(d), the paper raises several foundational issues that are particularly relevant for statistical induction as it relates to the current discussions on the replication crises and the trustworthiness of empirical evidence, arguing that: [i] The alleged Bernoulli swindle is grounded in the unwarranted claim \(\hat{\theta}_n (\mathbf{x}_0) \simeq \theta^* \), for a large enough \(n\), where \(\hat{\theta}_n (\mathbf{X})\) is an optimal estimator of the true value \(\theta^*\) of \(\theta \). [ii] Frequentist error probabilities are \textit{not} conditional on hypotheses (\(H_0\) and \(H_1\)) framed in terms of an unknown parameter \(\theta\) since \(\theta\) is neither a random variable nor an event. [iii] The direct versus inverse inference problem is a contrived and misplaced charge since neither conditional distribution \(f(\mathbf{x}_0 |\theta)\) and \(f(\theta |\mathbf{x}_0 )\) exists (formally or logically) in model-based \((\mathcal{M}_{\theta} (\mathbf{x}))\) frequentist inference.A strong duality principle for equivalence couplings and total variationhttps://zbmath.org/1528.600092024-03-13T18:33:02.981707Z"Jaffe, Adam Quinn"https://zbmath.org/authors/?q=ai:jaffe.adam-quinnSummary: We introduce and study a notion of duality for two classes of optimization problems commonly occurring in probability theory. That is, on an abstract measurable space \((\Omega, \mathcal{F})\), we consider pairs \((E, \mathcal{G})\) where \(E\) is an equivalence relation on \(\Omega\) and \(\mathcal{G}\) is a sub-\(\sigma\)-algebra of \(\mathcal{F}\); we say that \((E, \mathcal{G})\) satisfies ``strong duality'' if \(E\) is \((\mathcal{F} \otimes \mathcal{F})\)-measurable and if for all probability measures \(\mathbb{P}\), \(\mathbb{P}^\prime\) on \((\Omega, \mathcal{F})\) we have
\[
\max_{A \in \mathcal{G}} | \mathbb{P}(A) - \mathbb{P}^\prime(A) | = \min_{\widetilde{\mathbb{P}} \in \Pi (\mathbb{P}, \mathbb{P}^\prime)}(1 - \widetilde{\mathbb{P}}(E)),
\]
where \(\Pi(\mathbb{P}, \mathbb{P}^\prime)\) denotes the space of couplings of \(\mathbb{P}\) and \(\mathbb{P}^\prime\), and where ``max'' and ``min'' assert that the supremum and infimum are in fact achieved. The results herein give wide sufficient conditions for strong duality to hold, thereby extending a form of Kantorovich duality to a class of cost functions which are irregular from the point of view of topology but regular from the point of view of descriptive set theory. The given conditions recover or strengthen classical results, and they have novel consequences in stochastic calculus, point process theory, and random sequence simulation.Process convergence of fluctuations of linear eigenvalue statistics of random circulant matriceshttps://zbmath.org/1528.600102024-03-13T18:33:02.981707Z"Bose, Arup"https://zbmath.org/authors/?q=ai:bose.arup"Maurya, Shambhu Nath"https://zbmath.org/authors/?q=ai:maurya.shambhu-nath"Saha, Koushik"https://zbmath.org/authors/?q=ai:saha.koushikThe authors study the time dependent fluctuations of linear eigenvalue statistics for random circulant matrices. The study of the linear eigenvalue statistics of random matrices is one of the popular area of research in random matrix theory. In this direction, to be more precise, the process convergence of the time dependent fluctuations of linear eigenvalue statistics of random circulant matrices with independent Brownian motion entries is discussed, as the dimension of the matrix tends to infinity. The derivation is based on the trace formula of circulant matrix, the method of moments and some combinatorial techniques. All limiting processes are Gaussian.
Reviewer: Yuliya S. Mishura (Kyïv)Sine-kernel determinant on two large intervalshttps://zbmath.org/1528.600112024-03-13T18:33:02.981707Z"Fahs, Benjamin"https://zbmath.org/authors/?q=ai:fahs.benjamin"Krasovsky, Igor"https://zbmath.org/authors/?q=ai:krasovsky.igor-vSummary: We consider the probability of two large gaps (intervals without eigenvalues) in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We determine the multiplicative constant in the asymptotics. We also provide the full explicit asymptotics (up to decreasing terms) for the transition between one and two large gaps.
{\copyright} 2023 The Authors. \textit{Communications on Pure and Applied Mathematics} published by Courant Institute of Mathematics and Wiley Periodicals LLC.High-dimensional regime for Wishart matrices based on the increments of the solution to the stochastic heat equationhttps://zbmath.org/1528.600122024-03-13T18:33:02.981707Z"Gamain, Julie"https://zbmath.org/authors/?q=ai:gamain.julie"Mollinedo, David A. C."https://zbmath.org/authors/?q=ai:mollinedo.david-a-c"Tudor, Ciprian A."https://zbmath.org/authors/?q=ai:tudor.ciprian-aThe authors consider an \(n \times d\) random matrix \(\mathcal X_{n,d}\) whose entries are the spatial increments of the solution to the stochastic heat equation with spacetime white noise. They analyze the limit behavior of the associated Wishart matrix \(\frac 1d\mathcal X_{n,d} \mathcal X_{n,d}^\top\). They show that the Wishart matrix converges almost surely to a multiple of the identity matrix and that the suitably normalized Wishart matrix satisfies a central limit theorem. The proofs are based on the Wiener chaos decomposition, Malliavin calculus and Stein's method.
Reviewer: Zakhar Kabluchko (Münster)Uniform point variance bounds in classical beta ensembleshttps://zbmath.org/1528.600132024-03-13T18:33:02.981707Z"Najnudel, Joseph"https://zbmath.org/authors/?q=ai:najnudel.joseph"Virág, Bálint"https://zbmath.org/authors/?q=ai:virag.balintAuthors' summary: ``In this paper, we give bounds on the variance of the number of points of the Circular and the Gaussian \(\beta\) Ensemble in arcs of the unit circle or intervals of the real line. These bounds are logarithmic with respect to the renormalized length of these sets, which is expected to be optimal up to a multiplicative constant depending only on \(\beta \).''
For \(\beta > 0\), the following two ensembles are considered:
The Circular \(\beta\) Ensemble consists of \(n\) random points whose joint distribution has a density of the form
\[
Z_{n,\beta}^{-1}{\displaystyle \prod_{1 \leq j < k \leq n}} |\lambda_j - \lambda_k|^{\beta}
\]
with respect to Lebesgue measure on the \(n\)th power of the unit circle, and
the Gaussian \(\beta\) Ensemble consists of \(n\) points on the real line with the density (with respect to Lebesgue measure on \(\mathbb{R}^n\))
\[
(Z'_{n,\beta})^{-1} \exp(-\frac{\beta}{4} \sum_{1 \leq j \leq n} \lambda_j^2) {\displaystyle \prod_{1 \leq j < k \leq n}} |\lambda_j - \lambda_k|^{\beta}.
\]
The number of points of the Circular \(\beta\) Ensemble of order \(n\) in an arc \(I\) of the unit circle is shown to have a variance bounded by \(C_{\beta}\log(2 + n|I|)\), where \(|I|\) is the length of \(I\) and \(C_{\beta}\) depends only on \(\beta\). The Gaussian case is much more difficult and technically demanding.The empirical distribution of the points of the Gaussian \(\beta\) Ensemble, suitably scaled, tends to the Wigner semicircle distribution. The authors investigate the fluctuations of the number of points in an interval \(I\) with respect to the limiting distribution and show that the expected squared difference grows logarithmically with \(\sqrt n|I|\).
Reviewer: Göran Högnäs (Åbo)Distances between states and between predicateshttps://zbmath.org/1528.600142024-03-13T18:33:02.981707Z"Jacobs, Bart"https://zbmath.org/authors/?q=ai:jacobs.bart"Westerbaan, Abraham"https://zbmath.org/authors/?q=ai:westerbaan.abraham-a|westerbaan.abrahamSummary: This paper gives a systematic account of various metrics on probability distributions (states) and on predicates. These metrics are described in a uniform manner using the validity relation between states and predicates. The standard adjunction between convex sets (of states) and effect modules (of predicates) is restricted to convex complete metric spaces and directed complete effect modules. This adjunction is used in two state-and-effect triangles, for classical (discrete) probability and for quantum probability.A note on some critical thresholds of Bernoulli percolationhttps://zbmath.org/1528.600152024-03-13T18:33:02.981707Z"Tang, Pengfei"https://zbmath.org/authors/?q=ai:tang.pengfeiSummary: Consider Bernoulli bond percolation on a locally finite, connected graph \(G\) and let \(p_{\text{cut}}\) be the threshold corresponding to a ``first-moment method'' lower bound.
\textit{J. Kahn} [Electron. Commun. Probab. 8, 184--187 (2003; Zbl 1060.60096)] constructed a counter-example to Lyons' conjecture of \(p_{\text{cut}}={p_c}\) and proposed a modification. Here we give a positive answer to Kahn's modified question. The key observation is that in Kahn's modification, the new expectation quantity also appears in the differential inequality of one-arm events. This links the question to a lemma of \textit{H. Duminil-Copin} and \textit{V. Tassion} [Commun. Math. Phys. 343, No. 2, 725--745 (2016; Zbl 1342.82026)]. We also study some applications for Bernoulli percolation on periodic trees.Introduction to stochastic geometryhttps://zbmath.org/1528.600162024-03-13T18:33:02.981707Z"Hug, Daniel"https://zbmath.org/authors/?q=ai:hug.daniel"Reitzner, Matthias"https://zbmath.org/authors/?q=ai:reitzner.matthiasSummary: This chapter introduces some of the fundamental notions from stochastic geometry. Background information from convex geometry is provided as far as this is required for the applications to stochastic geometry.
First, the necessary definitions and concepts related to geometric point processes and from convex geometry are provided. These include Grassmann spaces and invariant measures, Hausdorff distance, parallel sets and intrinsic volumes, mixed volumes, area measures, geometric inequalities and their stability improvements. All these notions and related results will be used repeatedly in the present and in the subsequent chapters of the book.
Second, a variety of important models and problems from stochastic geometry will be reviewed. Among these are the Boolean model, random geometric graphs, intersection processes of (Poisson) processes of affine subspaces, random mosaics, and random polytopes. We state the most natural problems and point out important new results and directions of current research.
For the entire collection see [Zbl 1350.60005].The basic distributional theory for the product of zero mean correlated normal random variableshttps://zbmath.org/1528.600172024-03-13T18:33:02.981707Z"Gaunt, Robert E."https://zbmath.org/authors/?q=ai:gaunt.robert-edwardSummary: The product of two zero mean correlated normal random variables, and more generally the sum of independent copies of such random variables, has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to be recorded. This review paper fills this gap by providing the basic distributional theory for the sum of independent copies of the product of two zero mean correlated normal random variables. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a list of related distributions. We also review how the product of two zero mean correlated normal random variables arises naturally as a limiting distribution, with an example given for the distributional approximation of double Wiener-Itô integrals.
{{\copyright} 2022 The Author. \textit{Statistica Neerlandica} published by John Wiley \& Sons Ltd on behalf of Netherlands Society for Statistics and Operations Research.}On quantile-based asymmetric family of distributions: properties and inferencehttps://zbmath.org/1528.600182024-03-13T18:33:02.981707Z"Gijbels, Irène"https://zbmath.org/authors/?q=ai:gijbels.irene"Karim, Rezaul"https://zbmath.org/authors/?q=ai:karim.rezaul"Verhasselt, Anneleen"https://zbmath.org/authors/?q=ai:verhasselt.anneleenSummary: In this paper, we provide a detailed study of a general family of asymmetric densities. In the general framework, we establish expressions for important characteristics of the distributions and discuss estimation of the parameters via method-of-moments as well as maximum likelihood estimation. Asymptotic normality results for the estimators are provided. The results under the general framework are then applied to some specific examples of asymmetric densities. The use of the asymmetric densities is illustrated in a real-data analysis.
{{\copyright} 2019 The Authors. International Statistical Review {\copyright} 2019 International Statistical Institute}Response to the letter to the editor on ``On quantile-based asymmetric family of distributions: properties and inference''https://zbmath.org/1528.600192024-03-13T18:33:02.981707Z"Gijbels, Irène"https://zbmath.org/authors/?q=ai:gijbels.irene"Karim, Rezaul"https://zbmath.org/authors/?q=ai:karim.rezaul"Verhasselt, Anneleen"https://zbmath.org/authors/?q=ai:verhasselt.anneleenSummary: \textit{F. J. R. Alvarez} [Int. Stat. Rev. 88, No. 3, 793--796 (2020; Zbl 1528.60021)] points out an identification problem for the four-parameter family of two-piece asymmetric densities introduced by \textit{V. Nassiri} and \textit{I. Loris} [J. Appl. Stat. 40, No. 5, 1090--1105 (2013; Zbl 1514.62770)]. This implies that statistical inference for that family is problematic. Establishing probabilistic properties for this four-parameter family however still makes sense. For the three-parameter family, there is no identification problem. The main contribution in [the authors, ibid. 87, No. 3, 471--504 (2019; Zbl 1528.60018)] is to provide asymptotic results for maximum likelihood and method-of-moments estimators for members of the three-parameter quantile-based asymmetric family of distributions.
{{\copyright} 2020 International Statistical Institute}On recursions for moments of a compound random variable: an approach using an auxiliary counting random variablehttps://zbmath.org/1528.600202024-03-13T18:33:02.981707Z"Kim, Yoora"https://zbmath.org/authors/?q=ai:kim.yooraSummary: We present an identity on moments of a compound random variable by using an auxiliary counting random variable. Based on this identity, we develop a new recurrence formula for obtaining the raw and central moments of any order for a given compound random variable.Letter to the editor: ``On quantile-based asymmetric family of distributions: properties and inference''https://zbmath.org/1528.600212024-03-13T18:33:02.981707Z"Rubio Alvarez, Francisco J."https://zbmath.org/authors/?q=ai:alvarez.francisco-j-rubioSummary: We show that the family of asymmetric distributions studied in a recent publication in the International Statistical Review is equivalent to the family of two-piece distributions. Moreover, we show that the location-scale asymmetric family proposed in that publication is non-identifiable (overparameterised), and it coincides with the family of two-piece distributions after removing the redundant parameters.
{{\copyright} 2020 International Statistical Institute}
Comment to the paper [\textit{I. Gijbels} et al., Int. Stat. Rev. 87, No. 3, 471--504 (2019; Zbl 1528.60018)].Joint occupation times in an infinite interval for spectrally negative Lévy processes on the last exit timehttps://zbmath.org/1528.600222024-03-13T18:33:02.981707Z"Li, Yingqiu"https://zbmath.org/authors/?q=ai:li.yingqiu"Wei, Yushao"https://zbmath.org/authors/?q=ai:wei.yushao"Hu, Yangli"https://zbmath.org/authors/?q=ai:hu.yangliLévy processes are a kind of processes with stationary independent increments and play a very important role in many fields. A spectrally negative Lévy process (SNLP) is a Lévy process without positive jumps. With the Markov property and infinite divisibility, it plays an important role in the study of financial risks. The occupation time of an SNLP plays an important role in the study of bankruptcy problems. The authors of this article use the Poisson approach and perturbation approach to find some joint Laplace transforms of the last exit time by their joint occupation times over semiinfinite intervals \((-\infty,0)\) and \((0,\infty)\) by virtue of the associated scale functions.
Reviewer: Ze-Chun Hu (Chengdu)Gaussian limits for scheduled traffic with super-heavy tailed perturbationshttps://zbmath.org/1528.600232024-03-13T18:33:02.981707Z"Araman, Victor F."https://zbmath.org/authors/?q=ai:araman.victor-f"Glynn, Peter W."https://zbmath.org/authors/?q=ai:glynn.peter-wSummary: A scheduled arrival model is one in which the \(j\)th customer is scheduled to arrive at time \(jh\) but the customer actually arrives at time \(jh + \xi_j\), where the \(\xi_j\)'s are independent and identically distributed. It has previously been shown that the arrival counting process for scheduled traffic obeys a functional central limit theorem (FCLT) with fractional Brownian motion (fBM) with Hurst parameter \(H\in(0, 1/2)\) when the \(\xi_j\)'s have a Pareto-like tail with tail exponent lying in \((0, 1)\). Such limit processes exhibit less variability than Brownian motion, because the scheduling feature induces negative correlations in the arrival process. In this paper, we show that when the tail of the \(\xi_j\)'s has a super-heavy tail, the FCLT limit process is Brownian motion (i.e., \(H = 1/2\)), so that the heaviness of the tails eliminates any remaining negative correlations and generates a limit process with independent increments. We further study the case when the \(\xi_j\)'s have a Cauchy-like tail, and show that the limit process in this setting is a fBM with \(H = 0\). So, this paper shows that the entire range of fBMs with \(H\in[0, 1/2]\) are possible as limits of scheduled traffic.\(U\)-statistics on the spherical Poisson spacehttps://zbmath.org/1528.600242024-03-13T18:33:02.981707Z"Bourguin, Solesne"https://zbmath.org/authors/?q=ai:bourguin.solesne"Durastanti, Claudio"https://zbmath.org/authors/?q=ai:durastanti.claudio"Marinucci, Domenico"https://zbmath.org/authors/?q=ai:marinucci.domenico"Peccati, Giovanni"https://zbmath.org/authors/?q=ai:peccati.giovanniSummary: We review a recent stream of research on normal approximations for linear functionals and more general U-statistics of wavelets/needlets coefficients evaluated on a homogeneous spherical Poisson field. We show how, by exploiting results from [\textit{G. Peccati} and \textit{C. Zheng}, Electron. J. Probab. 15, Paper No. 48, 1487--1527 (2010; Zbl 1228.60031)] based on Malliavin calculus and Stein's method, it is possible to assess the rate of convergence to Gaussianity for a triangular array of statistics with growing dimensions. These results can be exploited in a number of statistical applications, such as spherical density estimations, searching for point sources, estimation of variance, and the spherical two-sample problem.
For the entire collection see [Zbl 1350.60005].Precise deviations for discrete ensembleshttps://zbmath.org/1528.600252024-03-13T18:33:02.981707Z"Chen, Wen Xuan"https://zbmath.org/authors/?q=ai:chen.wen-xuan"Gao, Fu Qing"https://zbmath.org/authors/?q=ai:gao.fuqingA discrete \(\beta\)-ensemble is the joint distribution of particles \(x = ({x_1},\dots,{x_N}) \in {Z^N}\) represented by \(d{P_{N,V,\beta }}(x) = \frac{1}{{{Z_{N,V,\beta}}}}{\left| {{\Delta _N}(x)} \right|^\beta}\prod\limits_{k = 1}^N {{e^{ - \frac{{N\beta }}{2}V({x_k})}}d{\delta _x}} \), where \(\beta > 0\), \(V:R \to R{\Delta _N}(x) = \prod\nolimits_{1 \leq i < j \leq N} {({x_i} - {x_j})},\, {Z_{N,V,\beta}}\beta = 2{e^{ - NV(x)}}\) is a continuous function called potential, \({\Delta _N}(x) = \prod\nolimits_{1 \leq i < j \leq N} {({x_i} - {x_j})}\) is the Vandermonde determinant and \({Z_{N,V,\beta }}\) is the normalization constant. The paper considers precise deviations for discrete ensembles. For the \(\beta = 2\) case, the authors first establish an asymptotic formula of the Christoffel-Darboux kernel of the discrete orthogonal polynomials on an infinite regular lattice with weight \({e^{ - NV(x)}}\), and then use the asymptotic formula to get the precise deviations of the extreme value for the corresponding ensemble.
Reviewer: Oleg K. Zakusilo (Kyïv)Azuma-Hoeffding bounds for a class of urn modelshttps://zbmath.org/1528.600262024-03-13T18:33:02.981707Z"Dasgupta, Amites"https://zbmath.org/authors/?q=ai:dasgupta.amitesSummary: We obtain Azuma-Hoeffding bounds, which are exponentially decreasing, for the probabilities of being away from the limit for a class of urn models. The method consists of relating the variables to certain linear combinations using eigenvectors of the replacement matrix, thus bringing in appropriate martingales. Some cases of repeated eigenvalues are also considered using cyclic vectors. Moreover, strong convergence of proportions is proved as an application of these bounds.Metastability from the large deviations point of view: a \(\varGamma\)-expansion of the level two large deviations rate functional of non-reversible finite-state Markov chainshttps://zbmath.org/1528.600272024-03-13T18:33:02.981707Z"Landim, C."https://zbmath.org/authors/?q=ai:landim.claudioLet \({(X^{(n)}_t : t\geq0)}\) be a sequence of continuous-time Markov chains on a fixed finite state space. Let \({\mathcal{J}_n}\) be the level two large deviations rate functional for \({X^{(n)}_t}\) as \({t\to\infty}.\) Under a natural hypothesis on the jump rates proposed by the author and \textit{J. Beltran} in [J. Stat. Phys. 149, No. 4, 598--618 (2012; Zbl 1260.82063)], it is proved that for some weights \({1\prec\theta^{(1)}_ n\prec\ldots\prec\theta^{(q)}_n}\) and functionals \({\mathcal{J}^{(0)}, \mathcal{J}^{(1)},\ldots, \mathcal{J}^{(q)}}\) the functional \({\theta^{(p)}_n \mathcal{J}_n}\)\ \ \(\Gamma\)-converges to \(\mathcal{J}^{(p)}\) for each \({1\leq p\leq q}.\) It can be written as
\[
\mathcal{J}_n=\mathcal{J}^{(0)}+\sum_{p=1}^q\frac{1}{\theta^{(p)}_ n}\mathcal{J}^{(p)}.
\]
The weights correspond to the time-scales at which the sequence of Markov chains \({X^{(n)}_t}\) exhibit a metastable behavior, and the zero level sets of the rate functionals \({\mathcal{J}^{(p)}}\) identify the metastable states. The proof of the main result relies on the previous work of the author in collaboration with \textit{T. Xu} [ALEA, Lat. Am. J. Probab. Math. Stat. 13, No. 2, 725--751 (2016; Zbl 1346.60125)], where the metastable behavior of \({X^{(n)}_t}\) has been investigated.
Reviewer: Ivan Podvigin (Novosibirsk)On a Spitzer-type law of large numbers for partial sums of independent and identically distributed random variables under sub-linear expectationshttps://zbmath.org/1528.600282024-03-13T18:33:02.981707Z"Wang, Miaomiao"https://zbmath.org/authors/?q=ai:wang.miaomiao"Wang, Min"https://zbmath.org/authors/?q=ai:wang.min.1|wang.min.15|wang.min|wang.min.2|wang.min.7"Wang, Xuejun"https://zbmath.org/authors/?q=ai:wang.xuejunThe aim of the current paper is to give necessary and sufficient conditions for the Spitzer-type law of large numbers for the maximum of partial sums of independent and identically distributed random variables under sub-linear expectations. The authors work under the framework of the sub-linear expectation space and extend some results from the classical probability space to sub-linear expectation spaces.
Reviewer: Ecaterina Sava-Huss (Innsbruck)The Malliavin-Stein method on the Poisson spacehttps://zbmath.org/1528.600292024-03-13T18:33:02.981707Z"Bourguin, Solesne"https://zbmath.org/authors/?q=ai:bourguin.solesne"Peccati, Giovanni"https://zbmath.org/authors/?q=ai:peccati.giovanniSummary: This chapter provides a detailed and unified discussion of a collection of recently introduced techniques, allowing one to establish limit theorems with explicit rates of convergence, by combining the Stein's and Chen-Stein methods with Malliavin calculus. Some results concerning multiple integrals are discussed in detail.
For the entire collection see [Zbl 1350.60005].On the characterization of exchangeable sequences through reverse-martingale empirical distributionshttps://zbmath.org/1528.600302024-03-13T18:33:02.981707Z"Bladt, Martin"https://zbmath.org/authors/?q=ai:bladt.martin"Shaiderman, Dimitry"https://zbmath.org/authors/?q=ai:shaiderman.dimitrySummary: It is a well-known fact that an exchangeable sequence has empirical distributions that form a reverse-martingale. This paper is devoted to the proof of the converse statement. As a byproduct of the proof for the binary case, we introduce and discuss the notion of two-coloring exchangeability.Generalized Wiener-Hermite integrals and rough non-Gaussian Ornstein-Uhlenbeck processhttps://zbmath.org/1528.600312024-03-13T18:33:02.981707Z"Assaad, Obayda"https://zbmath.org/authors/?q=ai:assaad.obayda"Diez, Charles-Phillipe"https://zbmath.org/authors/?q=ai:diez.charles-phillipe"Tudor, Ciprian A."https://zbmath.org/authors/?q=ai:tudor.ciprian-aHermite processes constitute a class of self-similar processes with stationary increments. They are characterized by their order (an integer \(q\geq 1\)) and their Hurst parameter (or self-similarity index) \(H\). The authors discuss several properties of the generalized Hermite processes, which are non-Gaussian self-similar processes with \(q\geq 2\) and with self-similarity indices belonging to the whole interval \((0, 1)\). The Rosenblatt process is included as a particular case. The representation of such processes as Wiener integrals with respect to the standard Hermite process is produced, with some useful identities involving fractional integrals and derivatives as a by-product. Then the authors make a first step to construct a calculus with respect to the generalized Hermite process. They define Wiener integrals with respect to these processes, and this allows to define an associated Ornstein-Uhlenbeck process, which is called a generalized Hermite Ornstein-Uhlenbeck process (GHOU process). It is established that the GHOU process is well defined, in Wiener or pathwise sense, depending upon the values of the parameters that appear in its expression. Some other properties for this process (probability distribution, sample paths) are studied and its behaviour when the drift parameter is small is described.
Reviewer: Yuliya S. Mishura (Kyïv)Effective filtering for slow-fast systems via Wong-Zakai approximationhttps://zbmath.org/1528.600322024-03-13T18:33:02.981707Z"Li, Haoyuan"https://zbmath.org/authors/?q=ai:li.haoyuan"Li, Wenlei"https://zbmath.org/authors/?q=ai:li.wenlei"Qu, Shiduo"https://zbmath.org/authors/?q=ai:qu.shiduo"Shi, Shaoyun"https://zbmath.org/authors/?q=ai:shi.shaoyunIn the present paper, the authors consider the filtering problem for the following slow-fast system
\[
\left\{ \begin{array}{lr} dX^{\varepsilon} =\frac{1}{\varepsilon} AX^{\varepsilon}dt+
\frac{1}{\varepsilon} f(X^{\varepsilon}, Y^{\varepsilon})dt+ +\frac{1}{\sqrt{\varepsilon}}q_1dw_1,&\quad \text{in}\, H_1, \\
dY^{\varepsilon} =BY^{\varepsilon}dt+ g(X^{\varepsilon}, Y^{\varepsilon})dt+ q_2dw_2,&\quad \text{in}\, H_2, \end{array}\right. \tag{1}
\]
where \(H_1\) and \(H_2\) are infinite separable Hilbert spaces, \(A, B\) are linear operators, \(f, g\) are nonlinear functions, parameter \(\varepsilon\) is sufficiently small, \(w_1\) and \(w_2\) are mutually independent standard real-valued Brownian motions.
Recently \textit{H. Qiao} [J. Math. Anal. Appl. 487, No. 2, Article ID 123979, 22 p. (2020; Zbl 1440.37057)] and \textit{H. Qiao} et al. [Nonlinearity 31, No. 10, 4649--4666 (2018; Zbl 1394.60037)] published results on filtering problems, and proved that the filtering on the corresponding slow manifold can approximate that of the original system (1).
The fact that Brownian motion is not differentiable stimulated people to develop a smoother approximation for the filtering of the considered stochastic problems. One of the well-known methods is Wong-Zakai approximation. The key of the technique is to replace white noise with colored noise, the systems driven by which have many advantages in numerical simulations. In this paper the authors use the Wong-Zakai approximation to investigate the filtering problem for (1), i.e., they consider the corresponding Wong-Zakai system as follows:
\[
\left\{ \begin{array}{lr} dU^{\mu,\varepsilon} =\frac{1}{\varepsilon} A U^{\mu,\varepsilon} dt+
\frac{1}{\varepsilon} f(U^{\mu,\varepsilon}, V^{\mu,\varepsilon} )dt+ +\frac{1}{\sqrt{\varepsilon}}q_1z_1^{\mu}dt,&\quad \text{in}\, H_1, \\
d V^{\mu,\varepsilon}=B V^{\mu,\varepsilon} dt+ g(U^{\mu,\varepsilon},V^{\mu,\varepsilon} )dt+ q_2 z_2^{\mu}dt,&\quad \text{in}\, H_2, \end{array}\right.\tag{2}
\]
where \(z_k^{\mu}\) satisfy \[ dz_k^{\mu}=-\frac{1}{\mu} z_k^{\mu}dt +\frac{1}{\mu}dw_k, k=1,2, \] and \(\mu\) is a small parameter.
The authors use the filtering of the smooth reduced system of (2) on the slow manifold to approximate that of (1) via random invariant manifold theory and Wong-Zakai approximation. They establish the existence of the random invariant manifold for the Wong-Zakai system (2), show the solution of the original system can be well approximated by that of a smooth reduced system on the obtained slow manifold, and state that the filtering of the reduced system can well approximate the one of the original system.
Reviewer: Mikhail P. Moklyachuk (Kyïv)On the optional and orthogonal decompositions of supermartingales and applicationshttps://zbmath.org/1528.600332024-03-13T18:33:02.981707Z"Berkaoui, Abdelkarem"https://zbmath.org/authors/?q=ai:berkaoui.abdelkaremA set \(\mathcal{Q}\) of probability measures is considered such that each measure in \(\mathcal{Q}\) is absolutely continuous with respect to a given probability measure \(\mathbb{P}\) and \(\mathcal{Q}\) contains at least one measure equivalent to \(\mathbb{P}\). A process \(X\) is called a \(\mathcal{Q}\)-supermartingale (resp. \(\mathcal{Q}\)-martingale) if it is a \(\mathbb{Q}\)-supermartingale (resp. \(\mathbb{Q}\)-martingale) for all \(\mathbb{Q} \in\mathcal{Q}\). Sufficient and necessary conditions are given for \(\mathcal{Q}\) such that any \(\mathcal{Q}\)-supermartingale can be decomposed into the sum of a \(\mathcal{Q}\)-martingale and a decreasing process. An orthogonal decomposition of \(\mathcal{Q}\)-supermartingales is also provided and the result is applied to the orthogonal decomposition of the polar cone of \(\mathcal{Q}\).
Reviewer: Ferenc Weisz (Budapest)One dimensional martingale rearrangement couplingshttps://zbmath.org/1528.600342024-03-13T18:33:02.981707Z"Jourdain, B."https://zbmath.org/authors/?q=ai:jourdain.benjamin"Margheriti, W."https://zbmath.org/authors/?q=ai:margheriti.williamLet \(\mu\) and \(\nu\) be probability measures on \(\mathbb{R}\) with finite moment of order \(\rho\) for some \(\rho\geq 1\). Denote by \(\Pi(\mu,\nu)\) the set of couplings of \(\mu\) and \(\nu\), that is, the set of measures on \(\mathbb{R}^2\) with marginals \(\mu\) and \(\nu\). Also, let \(\Pi^{\mathrm{M}}(\mu,\nu)\) denote the set of martingale couplings of \(\mu\) and \(\nu\), that is, the set of \(M\in \Pi(\mu,\nu)\) with \(\int_{\mathbb{R}}y\frac{M(\mathrm{d}x, \mathrm{d}y)}{\mu(\mathrm{d}x)}=x\) \(\mu\) a.e.
Fix a coupling \(\pi\in \Pi(\mu,\nu)\) satisfying \(\int_{\mathbb{R}}f(x)\mu(\mathrm{d}x)\leq \int_{\mathbb{R}}f(y)\nu(\mathrm{d}y)\) for any convex function \(f:\mathbb{R}\to\mathbb{R}\) and \(\int_{[a,\infty)}(x-\int_{\mathbb{R}}y\frac{\pi(\mathrm{d}x, \mathrm{d}y)}{\mu(\mathrm{d}x)})\mu(\mathrm{d}x)\leq 0\) for any \(a\in\mathbb{R}\). The latter condition is called the barycentre dispersion assumption. The authors provide a martingale coupling \(M\in \Pi^{\mathrm{M}}(\mu, \nu)\) such that \(\mathcal{AW}_\rho(\pi, M)=\inf_{M^\prime\in \Pi^{\mathrm{M}}(\mu, \nu)}\mathcal{AW}_\rho(\pi, M^\prime)\), where \(\mathcal{AW}_\rho\) is the so called adapted Wasserstein distance.
Finally, the authors discuss stability in the adapted Wassertein distance of the inverse transform martingale coupling with respect to the marginals.
Reviewer: Alexander Iksanov (Kyïv)On a certain martingale representation and the related Infinite dimensional moment problemhttps://zbmath.org/1528.600352024-03-13T18:33:02.981707Z"Tamura, Yuma"https://zbmath.org/authors/?q=ai:tamura.yumaSummary: It is well-known that any \(L^2\)-martingale with respect to a Brownian filtration is represented by a stochastic integral with respect to the Brownian motion. The theorem can be proven based on the fact that linear combinations of exponential martingales (of a specific type) are dense in the mentioned set. In this paper, the necessary and sufficient conditions for expressing martingales as true identities rather than approximations are considered, which turns out to be an infinite dimensional moment problem. A typical moment problem is given as follows: for real sequences \((\mu_i )_{i=0}^{\infty}\), find the necessary and sufficient conditions for the existence of a distribution whose support is a subset of \([0,\infty )\) and the \(i\)-th moments is \(\mu_i\). This is a fundamental problem in probability theory or integral theory that was first proposed around 1894, but it is still being studied as of 2023. In this paper, we point out that this problem is related to the above problem through chaos expansion, and give a proof using a version of the moment problem.An optimal transport-based characterization of convex orderhttps://zbmath.org/1528.600362024-03-13T18:33:02.981707Z"Wiesel, Johannes"https://zbmath.org/authors/?q=ai:wiesel.johannes-c-w"Zhang, Erica"https://zbmath.org/authors/?q=ai:zhang.ericaSummary: For probability measures \(\mu\), \(\nu\), and \(\rho\), define the cost functionals
\[
C(\mu, \rho) := \sup_{\pi\in\Pi(\mu, \rho)}\int\langle x, y\rangle \pi(\mathrm{d}x, \mathrm{d}y) \text{ and } C(\nu, \rho) := \sup_{\pi\in\Pi(\nu, \rho)}\int\langle x, y\rangle\pi(\mathrm{d}x, \mathrm{d}y),
\]
where \(\langle\cdot, \cdot \rangle\) denotes the scalar product and \(\Pi(\cdot, \cdot)\) is the set of couplings. We show that two probability measures \(\mu\) and \(\nu\) on \(\mathbb{R}^d\) with finite first moments are in convex order (i.e., \(\mu\preccurlyeq_c\nu)\) iff \(C(\mu, \rho) \leq C(\nu, \rho)\) holds for all probability measures \(\rho\) on \(\mathbb{R}^d\) with bounded support. This generalizes a result by Carlier. Our proof relies on a quantitative bound for the infimum of \(\int f\mathrm{d}\nu - \int f\mathrm{d}\mu\) over all 1-Lipschitz functions \(f\), which is obtained through optimal transport (OT) duality and the characterization result of OT (couplings) by Rüschendorf, by Rachev, and by Brenier. Building on this result, we derive new proofs of well known one-dimensional characterizations of convex order. We also describe new computational methods for investigating convex order and applications to model-independent arbitrage strategies in mathematical finance.Description of an ecological niche for a mixed local/nonlocal dispersal: an evolution equation and a new Neumann condition arising from the superposition of Brownian and Lévy processeshttps://zbmath.org/1528.600372024-03-13T18:33:02.981707Z"Dipierro, Serena"https://zbmath.org/authors/?q=ai:dipierro.serena"Valdinoci, Enrico"https://zbmath.org/authors/?q=ai:valdinoci.enricoSummary: We propose here a motivation for a mixed local/nonlocal problem with a new type of Neumann condition.
Our description is based on formal expansions and approximations. In a nutshell, a biological species is supposed to diffuse either by a random walk or by a jump process, according to prescribed probabilities. If the process makes an individual exit the niche, it must come to the niche right away, by selecting the return point according to the underlying stochastic process. More precisely, if the random particle exits the domain, it is forced to immediately reenter the domain, and the new point in the domain is chosen randomly by following a bouncing process with the same distribution as the original one.
By a suitable definition outside the niche, the density of the population ends up solving a mixed local/nonlocal equation, in which the dispersion is given by the superposition of the classical and the fractional Laplacian. This density function satisfies two types of Neumann conditions, namely the classical Neumann condition on the boundary of the niche, and a nonlocal Neumann condition in the exterior of the niche.Some arithmetic properties of Pólya's urnhttps://zbmath.org/1528.600382024-03-13T18:33:02.981707Z"Fernández, José L."https://zbmath.org/authors/?q=ai:fernandez-muniz.jose-luis|fernandez-perez.jose-luis"Fernández, Pablo"https://zbmath.org/authors/?q=ai:fernandez-gallardo.pablo|fernandez.pablo-garciaSummary: Following \textit{A. W. Hales} [IEEE Trans. Inf. Theory 64, No. 4, Part 2, 3150--3152 (2018; Zbl 1391.60099)], the evolution of Pólya's urn may be interpreted as a walk, a Pólya walk, on the integer lattice \(\mathbb{N}^2\). We study the visibility properties of Pólya's walk or, equivalently, the divisibility properties of the composition of the urn. In particular, we are interested in the asymptotic average time that a Pólya walk is visible from the origin, or, alternatively, in the asymptotic proportion of draws so that the resulting composition of the urn is coprime. Via de Finetti's exchangeability theorem, Pólya's walk appears as a mixture of standard random walks. This paper is a follow-up of \textit{J. Cilleruelo} et al. [Eur. J. Comb. 75, 92--112 (2019; Zbl 1400.05217)], where similar questions were studied for standard random walks.John's walkhttps://zbmath.org/1528.600392024-03-13T18:33:02.981707Z"Gustafson, Adam"https://zbmath.org/authors/?q=ai:gustafson.adam"Narayanan, Hariharan"https://zbmath.org/authors/?q=ai:narayanan.hariharan.1|narayanan.hariharanThe authors introduce and investigate an affine-invariant random walk whose steps are generated with the help of uniform sampling from John's maximum-volume ellipsoids of a small radius of appropriately symmetrized convex bodies. Assuming that the radius of the John's ellipsoids is \(\mathrm{const}\,n^{-5/2}\) and that the initial distribution of the walk has a bounded density, it is proved that the walk mixes to achieve a total variation distance \(\varepsilon\) in \(O(n^7\log(1/\varepsilon))\) steps. Also, with the same assumption concerning the radius, it is shown that the walk achieves a polynomial mixing bound when started from an appropriately chosen point of the convex body.
Reviewer: Alexander Iksanov (Kyïv)Noise reinforced Lévy processes: Lévy-Itô decomposition and applicationshttps://zbmath.org/1528.600402024-03-13T18:33:02.981707Z"Rosales-Ortiz, Alejandro"https://zbmath.org/authors/?q=ai:rosales-ortiz.alejandroSummary: A step reinforced random walk is a discrete time process with memory such that at each time step, with fixed probability \(p \in(0, 1)\), it repeats a previously performed step chosen uniformly at random while with complementary probability \(1 - p\), it performs an independent step with fixed law. In the continuum, the main result of \textit{J. Bertoin} [Ann. Inst. Henri Poincaré, Probab. Stat. 56, No. 3, 2236--2252 (2020; Zbl 1477.60069)] states that the random walk constructed from the discrete-time skeleton of a Lévy process for a time partition of mesh-size \(1 \slash n\) converges, as \(n \uparrow \infty\) in the sense of finite dimensional distributions, to a process \(\hat{\xi}\) referred to as a noise reinforced Lévy process. Our first main result states that a noise reinforced Lévy process has rcll paths and satisfies a \(\text{noise reinforced}\) Lévy-Itô decomposition in terms of the \textit{noise reinforced} Poisson point process of its jumps. We introduce the joint distribution of a Lévy process and its reinforced version \((\xi, \hat{\xi})\) and show that the pair, conformed by the skeleton of the Lévy process and its step reinforced version, converge towards \((\xi, \hat{\xi})\) as the mesh size tend to 0. As an application, we analyse the rate of growth of \(\hat{\xi}\) at the origin and identify its main features as an infinitely divisible process.Hölder continuity of the convex minorant of a Lévy processhttps://zbmath.org/1528.600412024-03-13T18:33:02.981707Z"González Cázares, Jorge"https://zbmath.org/authors/?q=ai:gonzalez-cazares.jorge"Kramer-Bang, David"https://zbmath.org/authors/?q=ai:kramer-bang.david"Mijatović, Aleksandar"https://zbmath.org/authors/?q=ai:mijatovic.aleksandarSummary: We characterise the Hölder continuity of the convex minorant of most Lévy processes. The proof is based on a novel connection between the path properties of the Lévy process at zero and the boundedness of the set of \(r\)-slopes of the convex minorant.On single-layer potentials, pseudo-gradients and a jump theorem for an isotropic \(\alpha\)-stable stochastic processhttps://zbmath.org/1528.600422024-03-13T18:33:02.981707Z"Mamalyha, Khrystyna"https://zbmath.org/authors/?q=ai:mamalyha.khrystyna"Osypchuk, Mykhailo"https://zbmath.org/authors/?q=ai:osypchuk.m-mSummary: The aim of this paper is a behavior investigation of pseudo-gradients with respect to the spatial variable of a single-layer potential if the spatial point tends to some point in the carrier surface of the potential. The potentials connect to the generator of an isotropic \(\alpha\)-stable stochastic process with the power \(\alpha \in (1,2]\). This generator is the fractional Laplacian of the order \(\alpha\). Pseudo-gradient or fractional gradient is the pseudo-differential operator of the gradient type. Its order \(\beta\) is a positive number less than \(\alpha\). The jump theorem is known in the case of \(\beta =\alpha -1\). We present here the corresponding results in the cases of \(\beta <\alpha -1\) and \(\beta >\alpha -1\). In the first case there are no jumps, and in the second case there are no finite limits of the fractional gradients of the single-layer potential, when the spatial argument tends to some point placed on the potential carrier.Concatenation of nonhonest Feller processes, exit laws, and limit theorems on graphshttps://zbmath.org/1528.600432024-03-13T18:33:02.981707Z"Bobrowski, Adam"https://zbmath.org/authors/?q=ai:bobrowski.adamSummary: We provide a rather explicit formula for the resolvent of a concatenation of \(N\) processes in terms of their exit laws and certain probability measures characterizing the way the processes are concatenated. As an application, we prove an averaging principle saying that by concatenating asymptotically splittable processes one can approximate Markov chains.The affine ensemble: determinantal point processes associated with the \(ax + b\) grouphttps://zbmath.org/1528.600442024-03-13T18:33:02.981707Z"Abreu, Luís Daniel"https://zbmath.org/authors/?q=ai:abreu.luis-daniel"Balazs, Peter"https://zbmath.org/authors/?q=ai:balazs.peter.2"Jakšić, Smiljana"https://zbmath.org/authors/?q=ai:jaksic.smiljanaSummary: We introduce the affine ensemble, a class of determinantal point processes (DPP) in the half-plane \(\mathbb{C}^+\) associated with the \(ax + b\) (affine) group, depending on an admissible Hardy function \(\psi\). We obtain the asymptotic behavior of the variance, the exact value of the asymptotic constant, and non-asymptotic upper and lower bounds for the variance on a compact set \(\Omega \subset \mathbb{C}^+\). As a special case one recovers the DPP related to the weighted Bergman kernel. When \(\psi\) is chosen within a finite family whose Fourier transform are Laguerre functions, we obtain the DPP associated to hyperbolic Landau levels, the eigenspaces of the finite spectrum of the Maass Laplacian with a magnetic field.Determinantal point processeshttps://zbmath.org/1528.600452024-03-13T18:33:02.981707Z"Decreusefond, Laurent"https://zbmath.org/authors/?q=ai:decreusefond.laurent"Flint, Ian"https://zbmath.org/authors/?q=ai:flint.ian"Privault, Nicolas"https://zbmath.org/authors/?q=ai:privault.nicolas"Torrisi, Giovanni Luca"https://zbmath.org/authors/?q=ai:torrisi.giovanni-lucaSummary: In this survey we review two topics concerning determinantal (or fermion) point processes. First, we provide the construction of diffusion processes on the space of configurations whose invariant measure is the law of a determinantal point process. Second, we present some algorithms to sample from the law of a determinantal point process on a finite window. Related open problems are listed.
For the entire collection see [Zbl 1350.60005].\(U\)-statistics in stochastic geometryhttps://zbmath.org/1528.600462024-03-13T18:33:02.981707Z"Lachièze-Rey, Raphaël"https://zbmath.org/authors/?q=ai:lachieze-rey.raphael"Reitzner, Matthias"https://zbmath.org/authors/?q=ai:reitzner.matthiasSummary: A \(U\)-statistic of order \(k\) with kernel \(f: \mathbb{X}^k \rightarrow \mathbb{R}^d\) over a Poisson process \(\eta\) is defined as
\[
\displaystyle{\sum_{(x_1,\ldots,x_k)}f(x_1,\ldots,x_k),}
\]
where the summation is over \(k\)-tuples of distinct points of \(\eta\), under appropriate integrability assumptions on \(f\). \(U\)-statistics play an important role in stochastic geometry since many interesting functionals can be written as \(U\)-statistics, like intrinsic volumes of intersection processes, characteristics of random geometric graphs, volumes of random simplices, and many others. It turns out that the Wiener-Ito chaos expansion of a \(U\)-statistic is finite and thus Malliavin calculus is a particularly suitable method. Variance estimates, approximation of the covariance structure, and limit theorems which have been out of reach for many years can be derived. In this chapter we state the fundamental properties of \(U\)-statistics and investigate moment formulae. The main object of the chapter is to introduce the available limit theorems.
For the entire collection see [Zbl 1350.60005].Stochastic analysis for Poisson processeshttps://zbmath.org/1528.600472024-03-13T18:33:02.981707Z"Last, Günter"https://zbmath.org/authors/?q=ai:last.gunterSummary: This chapter develops some basic theory for the stochastic analysis of Poisson process on a general \(\sigma\)-finite measure space. After giving some fundamental definitions and properties (as the multivariate Mecke equation) the chapter presents the Fock space representation of square-integrable functions of a Poisson process in terms of iterated difference operators. This is followed by the introduction of multivariate stochastic Wiener-Itô integrals and the discussion of their basic properties. The chapter then proceeds with proving the chaos expansion of square-integrable Poisson functionals, and defining and discussing Malliavin operators. Further topics are products of Wiener-Itô integrals and Mehler's formula for the inverse of the Ornstein-Uhlenbeck generator based on a dynamic thinning procedure. The chapter concludes with covariance identities, the Poincaré inequality, and the FKG-inequality.
For the entire collection see [Zbl 1350.60005].Variational analysis of Poisson processeshttps://zbmath.org/1528.600482024-03-13T18:33:02.981707Z"Molchanov, Ilya"https://zbmath.org/authors/?q=ai:molchanov.ilya-s"Zuyev, Sergei"https://zbmath.org/authors/?q=ai:zuyev.sergeiSummary: The expected value of a functional \(F(\eta)\) of a Poisson process \(\eta\) can be considered as a function of its intensity measure \(\mu\). The paper surveys several results concerning differentiability properties of this functional on the space of signed measures with finite total variation. Then, necessary conditions for \(\mu\) being a local minima of the considered functional are elaborated taking into account possible constraints on \(\mu\), most importantly the case of \(\mu\) with given total mass \(a\). These necessary conditions can be phrased by requiring that the gradient of the functional (being the expected first difference \(F(\eta +\delta_x) - F(\eta))\) is constant on the support of \(\mu\). In many important cases, the gradient depends only on the local structure of \(\mu\) in a neighbourhood of \(x\) and so it is possible to work out the asymptotics of the minimising measure with the total mass \(a\) growing to infinity. Examples include the optimal approximation of convex functions, clustering problem and optimal search. In non-asymptotic cases, it is in general possible to find the optimal measure using steepest descent algorithms which are based on the obtained explicit form of the gradient.
For the entire collection see [Zbl 1350.60005].Extremal behavior of large cells in the Poisson hyperplane mosaichttps://zbmath.org/1528.600492024-03-13T18:33:02.981707Z"Otto, Moritz"https://zbmath.org/authors/?q=ai:otto.moritzSummary: We study the asymptotic behavior of a size-marked point process of centers of large cells in a stationary and isotropic Poisson hyperplane mosaic in dimension \(d \geq 2\). The sizes of the cells are measured by their inradius or their \(k\)th intrinsic volume \((k \geq 2)\), for example. We prove a Poisson limit theorem for this process in Kantorovich-Rubinstein distance and thereby generalize a result in
[\textit{N. Chenavier} and \textit{R. Hemsley}, Adv. Appl. Probab. 48, No. 2, 544--573 (2016; Zbl 1342.60011)] in various directions. Our proof is based on a general Poisson process approximation result that extends a theorem in
[\textit{O. Bobrowski} et al., Ann. Henri Lebesgue 5, 1489--1534 (2022; Zbl 1506.60019)].Poisson point process convergence and extreme values in stochastic geometryhttps://zbmath.org/1528.600502024-03-13T18:33:02.981707Z"Schulte, Matthias"https://zbmath.org/authors/?q=ai:schulte.matthias.1|schulte.matthias"Thäle, Christoph"https://zbmath.org/authors/?q=ai:thale.christophSummary: Let \(\eta_t\) be a Poisson point process with intensity measure \(t \mu\), \(t > 0\), over a Borel space \(\mathbb{X}\), where \(\mu\) is a fixed measure. Another point process \(\xi_t\) on the real line is constructed by applying a symmetric function \(f\) to every \(k\)-tuple of distinct points of \(\eta_t\). It is shown that \(\xi_t\) behaves after appropriate rescaling like a Poisson point process, as \(t \rightarrow \infty\), under suitable conditions on \(\eta_t\) and \(f\). This also implies Weibull limit theorems for related extreme values. The result is then applied to investigate problems arising in stochastic geometry, including small cells in Voronoi tessellations, random simplices generated by non-stationary hyperplane processes, triangular counts with angular constraints, and non-intersecting \(k\)-flats. Similar results are derived if the underlying Poisson point process is replaced by a binomial point process.
For the entire collection see [Zbl 1350.60005].On convergence of volume of level sets of stationary smooth Gaussian fieldshttps://zbmath.org/1528.600512024-03-13T18:33:02.981707Z"Beliaev, Dmitry"https://zbmath.org/authors/?q=ai:beliaev.dmitri-b"Hegde, Akshay"https://zbmath.org/authors/?q=ai:hegde.akshaySummary: We prove convergence of Hausdorff measure of level sets of smooth Gaussian fields when the levels converge. Given two coupled stationary fields \(f_1\), \(f_2\), we estimate the difference of Hausdorff measure of level sets in expectation, in terms of \(C^2\)-fluctuations of the field \(F = f_1 - f_2\). The main idea in the proof is to represent difference in volume as an integral of mean curvature using the divergence theorem. This approach is different from using Kac-Rice type formula as main tool in the analysis.Rudin extension theorems on product spaces, turning bands, and random fields on balls cross timehttps://zbmath.org/1528.600522024-03-13T18:33:02.981707Z"Porcu, Emilio"https://zbmath.org/authors/?q=ai:porcu.emilio"Feng, Samuel F."https://zbmath.org/authors/?q=ai:feng.samuel-f"Emery, Xavier"https://zbmath.org/authors/?q=ai:emery.xavier"Peron, Ana P."https://zbmath.org/authors/?q=ai:peron.ana-paulaSummary: Characteristic functions that are radially symmetric have a dual interpretation, as they can be used as the isotropic correlation functions of spatial random fields. Extensions of isotropic correlation functions from balls into \(d\)-dimensional Euclidean spaces, \(\mathbb{R}^d\), have been understood after Rudin. Yet, extension theorems on product spaces are elusive, and a counterexample provided by Rudin on rectangles suggests that the problem is challenging. This paper provides extension theorems for multiradial characteristic functions that are defined in balls embedded in \(\mathbb{R}^d\) cross, either \(\mathbb{R}^{d'}\) or the unit sphere \(\mathbb{S}^{d'}\) embedded in \(\mathbb{R}^{d'+1}\), for any two positive integers \(d\) and \(d'\). We then examine Turning Bands operators that provide bijections between the class of multiradial correlation functions in given product spaces, and multiradial correlations in product spaces having different dimensions. The combination of extension theorems with Turning Bands provides a connection with random fields that are defined in balls cross linear or circular time.On the capacity for degenerated \(G\)-Brownian motion and its applicationhttps://zbmath.org/1528.600532024-03-13T18:33:02.981707Z"Li, Xiaojuan"https://zbmath.org/authors/?q=ai:li.xiaojuan"Li, Xinpeng"https://zbmath.org/authors/?q=ai:li.xinpengSummary: In this paper, the capacity \(c(\{B_T \in A \})\) of degenerated \(G\)-Brownian motion \(B_T\) for arbitrary Borel set \(A\) is calculated via the viscosity solution of \(G\)-heat equation. In particular, unlike the classical Brownian motion and non-degenerated \(G\)-Brownian motion which do not weight points, we obtain \(c(\{B_T = a \}) > 0\) for each \(a \in \mathbb{R}\) in the degenerated case we study here. As an application, we show that \(I_A(B_T)\) has no quasi-continuous version unless \(A = \emptyset\) or \(\mathbb{R}\).Universality of Poisson limits for moduli of roots of Kac polynomialshttps://zbmath.org/1528.600542024-03-13T18:33:02.981707Z"Cook, Nicholas A."https://zbmath.org/authors/?q=ai:cook.nicholas-a"Nguyen, Hoi H."https://zbmath.org/authors/?q=ai:nguyen.hoi-h"Yakir, Oren"https://zbmath.org/authors/?q=ai:yakir.oren"Zeitouni, Ofer"https://zbmath.org/authors/?q=ai:zeitouni.oferSummary: We give a new proof of a recent resolution [``Random polynomials: the closest roots to the unit circle'', Preprint, \url{arXiv:2010.1086918}] by \textit{M. Michelen} and \textit{J. Sahasrabudhe} of a conjecture of \textit{L. A. Shepp} and \textit{R. J. Vanderbei} [Trans. Am. Math. Soc. 347, No. 11, 4365--4384 (1995; Zbl 0841.30006)] that the moduli of roots of Gaussian Kac polynomials of degree \(n\), centered at \(1\) and rescaled by \(n^2\), should form a Poisson point process. We use this new approach to verify a conjecture from [Michelen and Sahasrabudhe, loc. cit.] that the Poisson statistics are in fact universal.Combinatorics of Poisson stochastic integrals with random integrandshttps://zbmath.org/1528.600552024-03-13T18:33:02.981707Z"Privault, Nicolas"https://zbmath.org/authors/?q=ai:privault.nicolasSummary: We present a self-contained account of recent results on moment identities for Poisson stochastic integrals with random integrands, based on the use of functional transforms on the Poisson space. This presentation relies on elementary combinatorics based on the Faà di Bruno formula, partitions and polynomials, which are used together with multiple stochastic integrals, finite difference operators and integration by parts.
For the entire collection see [Zbl 1350.60005].Malliavin calculus for stochastic processes and random measures with independent incrementshttps://zbmath.org/1528.600562024-03-13T18:33:02.981707Z"Solé, Josep Lluís"https://zbmath.org/authors/?q=ai:sole.josep-lluis"Utzet, Frederic"https://zbmath.org/authors/?q=ai:utzet.fredericSummary: Malliavin calculus for Poisson processes based on the \textit{difference operator} or \textit{add-one-cost operator} is extended to stochastic processes and random measures with independent increments. Our approach is to use a Wiener-Itô chaos expansion, valid for both stochastic processes and random measures with independent increments, to construct a Malliavin derivative and a Skorohod integral. Useful derivation rules for smooth functionals given by
\textit{C. Geiss} and \textit{E. Laukkarinen} [Probab. Math. Stat. 31, No. 1, 1--15 (2011; Zbl 1260.60105)] are proved. In addition, characterizations for processes or random measures with independent increments based on the duality between the Malliavin derivative and the Skorohod integral following an interesting point of view from [\textit{R. Murr}, Stochastic Processes Appl. 123, No. 5, 1729--1749 (2013; Zbl 1274.60150)] are studied.
For the entire collection see [Zbl 1350.60005].A new hybrid approach for nonlinear stochastic differential equations driven by multifractional Gaussian noisehttps://zbmath.org/1528.600572024-03-13T18:33:02.981707Z"Eftekhari, Tahereh"https://zbmath.org/authors/?q=ai:eftekhari.tahereh"Rashidinia, Jalil"https://zbmath.org/authors/?q=ai:rashidinia.jalil(no abstract)Backward stochastic evolution inclusions in UMD Banach spaceshttps://zbmath.org/1528.600582024-03-13T18:33:02.981707Z"Essaky, E. H."https://zbmath.org/authors/?q=ai:essaky.el-hassan"Hassani, M."https://zbmath.org/authors/?q=ai:hassani.mohammed"Rhazlane, C. E."https://zbmath.org/authors/?q=ai:rhazlane.c-eSummary: In this paper, we prove the existence of a mild \(L^p\)-solution for the backward stochastic evolution inclusion (BSEI for short) of the form
\[
\begin{cases}
d Y_t + A Y_t dt \in G (t, Y_t, Z_t) dt + Z_t d W_t, \quad t \in [0, T] \\
Y_T = \xi,
\end{cases}
\]
where \(W= (W_t )_{t \in [0, T]}\) is a standard Brownian motion, \(A\) is the generator of a \(C_0\)-semigroup on a UMD Banach space \(E, \xi\) is a terminal condition from \(L^p (\Omega, \mathscr{F}_T; E)\), with \(p>1\) and \(G\) is a set-valued function satisfying some suitable conditions.
The case when the processes with values in spaces that have martingale type \(2\) has been also studied.Almost surely exponential stability of semi-Markovian switched singular stochastic systems with mode-dependent rankshttps://zbmath.org/1528.600592024-03-13T18:33:02.981707Z"Hu, Zenghui"https://zbmath.org/authors/?q=ai:hu.zenghui"Mu, Xiaowu"https://zbmath.org/authors/?q=ai:mu.xiaowu(no abstract)Existence of solution for Volterra-Fredholm type stochastic fractional integro-differential system of order \(\mu \in (1, 2)\) with sectorial operatorshttps://zbmath.org/1528.600602024-03-13T18:33:02.981707Z"Kaliraj, K."https://zbmath.org/authors/?q=ai:kaliraj.kalimuthu"Muthuvel, K."https://zbmath.org/authors/?q=ai:muthuvel.kandasamySummary: The mainspring of the study is to investigate the out-turn of stochastic Volterra-Fredholm integro-differential inclusion of order \(\mu \in \left(1,2\right)\) with sectorial operator of the type \(\left(P,\eta, \varrho, \gamma \right)\). The existence results of our proposed problem is derived by employing Martelli's fixed point approach. We do not limit the theoretical results of fractional stochastic equation to local condition but extend to nonlocal condition, and physical interpretation of our obtained results is proved with an appropriate illustration.
{{\copyright} 2023 John Wiley \& Sons, Ltd.}An approach to solving input reconstruction problems in stochastic differential equations: dynamic algorithms and tuning their parametershttps://zbmath.org/1528.600612024-03-13T18:33:02.981707Z"Rozenberg, Valeriy"https://zbmath.org/authors/?q=ai:rozenberg.valerii-lvovichSummary: Within the framework of the key approach from the theory of dynamic inversion, input reconstruction problems for stochastic differential equations are investigated. Different types of input information are used for the simultaneous reconstruction of disturbances in both the deterministic and stochastic terms of the equations. Feasible solving algorithms are designed; estimates of their convergence rates are derived. An empirical procedure adapting an algorithm to a specific system's dynamics to obtain best approximation results is discussed. An illustrative example for this technique is presented.
For the entire collection see [Zbl 1517.90002].Random periodicity for stochastic Liénard equationshttps://zbmath.org/1528.600622024-03-13T18:33:02.981707Z"Uda, Kenneth"https://zbmath.org/authors/?q=ai:uda.kennethSummary: In this paper, we establish some sufficient conditions for the existence of random limit cycle generated by stochastic Liénard equation. Our technique involve Lyapunov functions and truncation arguments. Furthermore, using polar coordinate transformation and rigid rotation, we further established existence (non-existence) of a possible minimal period of random periodic solution of stochastic van der Pol oscillator.Sobolev space weak solutions to one kind of quasilinear parabolic partial differential equations related to forward-backward stochastic differential equationshttps://zbmath.org/1528.600632024-03-13T18:33:02.981707Z"Wu, Zhen"https://zbmath.org/authors/?q=ai:wu.zhen"Xie, Bing"https://zbmath.org/authors/?q=ai:xie.bing"Yu, Zhiyong"https://zbmath.org/authors/?q=ai:yu.zhiyongSummary: This paper is concerned with the Sobolev type weak solutions of one class of second order quasilinear parabolic partial differential equations (PDEs, for short). First of all, similar to \textit{C. Feng} et al. [J. Differ. Equations 264, No. 2, 959--1018 (2018; Zbl 1516.35184)] and \textit{Z. Wu} and \textit{Z. Yu} [Stochastic Processes Appl. 124, No. 12, 3921--3947 (2014; Zbl 1314.60135)], we use a family of coupled forward-backward stochastic differential equations (FBSDEs, for short) which satisfy the monotonous assumption to represent the classical solutions of the quasilinear PDEs. Then, based on the classical solutions of a family of PDEs approximating the weak solutions of the quasilinear PDEs, we prove the existence of the weak solutions. Moreover, the principle of norm equivalence is employed to link FBSDEs and PDEs to obtain the uniqueness of the weak solutions. In summary, we provide a probabilistic interpretation for the weak solutions of quasilinear PDEs, which enriches the theory of nonlinear PDEs.Fast-slow stochastic dynamical system with singular coefficientshttps://zbmath.org/1528.600642024-03-13T18:33:02.981707Z"Xie, Longjie"https://zbmath.org/authors/?q=ai:xie.longjieSummary: This paper aims to study the asymptotic behavior of a fast-slow stochastic dynamical system with singular coefficients, where the fast motion is given by a continuous diffusion process while the slow component is driven by an \(\alpha\)-stable noise with \(\alpha \in [1, 2)\). Using Zvonkin's transformation and the technique of the Poisson equation, we have that both the strong and weak convergences in the averaging principle are established, which can be viewed as a functional law of large numbers. Then we study the small fluctuations between the original system around its average. We show that the normalized difference converges weakly to an Ornstein-Uhlenbeck type Gaussian process, which is a form of the functional central limit theorem. Furthermore, sharp rates for the above convergences are also obtained, and these convergences are shown to not depend on the regularities of the coefficients with respect to the fast variable, which reflect the effects of noises on the multi-scale systems.Splitting scheme for backward doubly stochastic differential equationshttps://zbmath.org/1528.600652024-03-13T18:33:02.981707Z"Bao, Feng"https://zbmath.org/authors/?q=ai:bao.feng"Cao, Yanzhao"https://zbmath.org/authors/?q=ai:cao.yanzhao"Zhang, He"https://zbmath.org/authors/?q=ai:zhang.heThe authors introduce a numerical scheme for a class of backward doubly stochastic differential equations (BDSDEs). They decompose the backward doubly stochastic differential equation into a backward stochastic differential equation and a stochastic differential equation, which are easier to analyze than the original BDSDE itself. The two equations in the decomposition are approximated using first-order finite difference schemes, which results in a first-order scheme for the backward doubly stochastic differential equation. Numerical experiments are included to illustrate the proposed scheme.
Reviewer: Maria Gordina (Storrs)Analytical manner for abundant stochastic wave solutions of extended KdV equation with conformable differential operatorshttps://zbmath.org/1528.600662024-03-13T18:33:02.981707Z"Hyder, Abd-Allah"https://zbmath.org/authors/?q=ai:hyder.abdallah"Soliman, Ahmed H."https://zbmath.org/authors/?q=ai:soliman.ahmed-husseinSummary: This work investigates the Wick-type stochastic and extended Korteweg-de Vries (EKdV) equation under conformable differential operators. Novel wave solutions with soliton and periodic types are produced to the deterministic EKdV under conformable differential operators. Abundant stochastic wave solutions are explored for the Wick-type stochastic EKdV equation by using conformable differential operators and the inverse Hermite transform. The stochastic soliton and periodic solutions are denoted as functional solutions with Brownian motion environment. For the acquired solutions, comparisons and graphical representations with three-dimensional graphs are shown for peculiar values of the existing parameters. According to the existing literature, utilization of the conformable differential operators is a novel contribution for solving the stochastic EKDV and other stochastic nonlinear problems.
{{\copyright} 2021 John Wiley \& Sons, Ltd.}Space-time stochastic calculus and white noisehttps://zbmath.org/1528.600672024-03-13T18:33:02.981707Z"Øksendal, Bernt"https://zbmath.org/authors/?q=ai:oksendal.bernt-karstenSummary: In the first part of this paper I give the historical background to my initial interest in stochastic analysis and to the writing of my book [Stochastic differential equations. An introduction with applications. Berlin, etc.: Springer Verlag (1985; Zbl 0567.60055)]. The first edition of this book was published by Springer in 1985, with the highly appreciated support of Catriona Byrne.
In the second part I present a motivation for modelling the dynamics of a system subject to a noise by means of a stochastic partial differential equation (SPDE) driven by a time-space Brownian sheet. This is followed by a brief survey of time-space white noise and Hida-Malliavin calculus, which are useful tools for studying such equations.
As an illustration I apply white noise calculus to find an explicit solution of an SPDE describing population growth in an environment subject to time-space white noise.
For the entire collection see [Zbl 1515.01005].Averaging principle and normal deviations for multi-scale stochastic hyperbolic-parabolic equationshttps://zbmath.org/1528.600682024-03-13T18:33:02.981707Z"Röckner, Michael"https://zbmath.org/authors/?q=ai:rockner.michael"Xie, Longjie"https://zbmath.org/authors/?q=ai:xie.longjie"Yang, Li"https://zbmath.org/authors/?q=ai:yang.li.7The paper studies asymptotic behavior of stochastic hyperbolic-parabolic equations with slow-fast time scales, and establish both the strong and weak convergence in the averaging principle setting. The stochastic fluctuations of the original system around its averaged equation is investigated as well. It turns out that the normalized difference converges weakly to the solution of a linear stochastic wave equation. Also, sharp rates for the above convergence have been obtained.
Reviewer: Anatoliy Swishchuk (Calgary)A new type distribution-dependent SDE for singular nonlinear PDEhttps://zbmath.org/1528.600692024-03-13T18:33:02.981707Z"Wang, Feng-Yu"https://zbmath.org/authors/?q=ai:wang.feng-yuThe author studies a new type SDE depending on the future distributions with all initial values as follows:
\[
dX_{s,t}^{x}=\sigma_{t}(X_{s,t}^{x})dW_{t}+\{b_{T-t}+F_{T-t}(\cdot,\psi_{t},\nabla \psi_{t})\}(X_{s,t}^{x})dt,
\]
\[
\psi_{t}(y)=\mathbb{E}[u_{0}(X_{t,T}^{y}){\mathrm{e}}^{\int_{t}^{T}V_{T-r}(X_{t,r}^{y})dr}+\int_{t}^{T}g_{T-r}(X_{t,r}^{y}){\mathrm{e}}^{\int_{t}^{T}V_{T-r'}(X_{t,r'}^{y})dr'}dr],\quad y\in\mathbb{R}^{d}.
\]
The author establishes the correspondence between this equation and the associated singular nonlinear PDE. Well-posedness and regularities are investigated.
Reviewer: Feng Chen (Changchun)Existence, uniqueness, and averaging principle for Hadamard Itô-Doob stochastic delay fractional integral equationshttps://zbmath.org/1528.600702024-03-13T18:33:02.981707Z"Ben Makhlouf, Abdellatif"https://zbmath.org/authors/?q=ai:ben-makhlouf.abdellatif"Mchiri, Lassaad"https://zbmath.org/authors/?q=ai:mchiri.lassaad"Mtiri, Foued"https://zbmath.org/authors/?q=ai:mtiri.foued(no abstract)Propagation of minimality in the supercooled Stefan problemhttps://zbmath.org/1528.600712024-03-13T18:33:02.981707Z"Cuchiero, Christa"https://zbmath.org/authors/?q=ai:cuchiero.christa"Rigger, Stefan"https://zbmath.org/authors/?q=ai:rigger.stefan"Svaluto-Ferro, Sara"https://zbmath.org/authors/?q=ai:svaluto-ferro.saraSupercooled Stefan problems describe the evolution of the boundary between the solid and liquid phases of a substance, where the liquid is assumed to be cooled below its freezing point. Following the methodology of \textit{F. Delarue} et al. [Probab. Math. Phys. 3, No. 1, 171--213 (2022; Zbl 1489.35309)], the authors construct solutions to the one-phase one-dimensional supercooled Stefan problem through a certain McKean-Vlasov equation, which allows to define global solutions even in the presence of blow-ups. Solutions to the McKean-Vlasov equation arise as mean-field limits of particle systems interacting through hitting times, which is important for systemic risk modeling. The main contributions of this paper are: (i) A general tightness theorem for the Skorokhod M1-topology which applies to processes that can be decomposed into a continuous and a monotone part. (ii) A propagation of chaos result for a perturbed version of the particle system for general initial conditions. (iii) The proof of a conjecture of Delarue, Nadtochiy and Shkolnikov [loc. cit.], relating the solution concepts of so-called minimal and physical solutions, showing that minimal solutions of the McKean-Vlasov equation are physical whenever the initial condition is integrable.
Reviewer: Hossam A. Ghany (al-Qāhira)Quantized noncommutative Riemann manifolds and stochastic processes: the theoretical foundations of the square root of Brownian motionhttps://zbmath.org/1528.600722024-03-13T18:33:02.981707Z"Frasca, Marco"https://zbmath.org/authors/?q=ai:frasca.marco"Farina, Alfonso"https://zbmath.org/authors/?q=ai:farina.alfonso"Alghalith, Moawia"https://zbmath.org/authors/?q=ai:alghalith.moawiaSummary: We lay the theoretical and mathematical foundations of the square root of Brownian motion and we prove the existence of such a process. In doing so, we consider Brownian motion on quantized noncommutative Riemannian manifolds and show how a set of stochastic processes on sets of complex numbers can be devised. This class of stochastic processes are shown to yield at the outset a Chapman-Kolmogorov equation with a complex diffusion coefficient that can be straightforwardly reduced to the Schrödinger equation. The existence of these processes has been recently shown numerically. In this work we provide an analogous support for the existence of the Chapman-Kolmogorov-Schrödinger equation for them, performing a Monte Carlo study. It is numerically seen as a Wick rotation can turn the heat kernel into the Schrödinger one, mapping such kernels through the corresponding stochastic processes. In this way, we introduce a new kind of improper complex stochastic process. This permits a reformulation of quantum mechanics using purely geometrical concepts that are strongly linked to stochastic processes. Applications to economics are also entailed.Approximation of the invariant measure of stable SDEs by an Euler-Maruyama schemehttps://zbmath.org/1528.600732024-03-13T18:33:02.981707Z"Chen, Peng"https://zbmath.org/authors/?q=ai:chen.peng.3"Deng, Chang-Song"https://zbmath.org/authors/?q=ai:deng.changsong"Schilling, René L."https://zbmath.org/authors/?q=ai:schilling.rene-leander"Xu, Lihu"https://zbmath.org/authors/?q=ai:xu.lihuSummary: We propose two Euler-Maruyama (EM) type numerical schemes in order to approximate the invariant measure of a stochastic differential equation (SDE) driven by an \(\alpha\)-stable Lévy process \((1<\alpha <2)\): an approximation scheme with the \(\alpha\)-stable distributed noise and a further scheme with Pareto-distributed noise. Using a discrete version of Duhamel's principle and Bismut's formula in Malliavin calculus, we prove that the error bounds in Wasserstein-1 distance are in the order of \(\eta^{1-\epsilon}\) and \(\eta^{\frac{2}{\alpha}-1}\), respectively, where \(\epsilon \in (0,1)\) is arbitrary and \(\eta\) is the step size of the approximation schemes. For the Pareto-driven scheme, an explicit calculation for Ornstein-Uhlenbeck \(\alpha\)-stable process shows that the rate \(\eta^{\frac{2}{\alpha}-1}\) cannot be improved.Regularisation by fractional noise for one-dimensional differential equations with distributional drifthttps://zbmath.org/1528.600742024-03-13T18:33:02.981707Z"Anzeletti, Lukas"https://zbmath.org/authors/?q=ai:anzeletti.lukas"Richard, Alexandre"https://zbmath.org/authors/?q=ai:richard.alexandre"Tanré, Etienne"https://zbmath.org/authors/?q=ai:tanre.etienneSummary: We study existence and uniqueness of solutions to the equation \(d X_t = b(X_t) d t + d B_t\), where \(b\) is a distribution in some Besov space and \(B\) is a fractional Brownian motion with Hurst parameter \(H \leqslant 1 \slash 2\). First, the equation is understood as a nonlinear Young equation. This involves a nonlinear Young integral constructed in the space of functions with finite \(p\)-variation, which is well suited when \(b\) is a measure. Depending on \(H\), a condition on the Besov regularity of \(b\) is given so that solutions to the equation exist. The construction is deterministic, and \(B\) can be replaced by a deterministic path \(w\) with a sufficiently smooth local time. Using this construction we prove the existence of weak solutions (in the probabilistic sense).
We also prove that solutions coincide with limits of strong solutions obtained by regularisation of \(b\). This is used to establish pathwise uniqueness and existence of a strong solution. In particular when \(b\) is a finite measure, weak solutions exist for \(H < \sqrt{2} - 1\), while pathwise uniqueness and strong existence hold when \(H \leqslant 1 \slash 4\). The proofs involve fine properties of the local time of the fractional Brownian motion, as well as new regularising properties of this process which are established using the stochastic sewing Lemma.Rates of convergence for Gibbs sampling in the analysis of almost exchangeable datahttps://zbmath.org/1528.600752024-03-13T18:33:02.981707Z"Gerencsér, Balázs"https://zbmath.org/authors/?q=ai:gerencser.balazs"Ottolini, Andrea"https://zbmath.org/authors/?q=ai:ottolini.andreaThe paper analyses a Gibbs sampler for sampling \(\mathbf p = (p_1, ..., p_d) \in [0,1]^d\) from a distribution with density proportional to
\[
\exp\Bigl( - A^2 \sum_{i,j \in [d] : i < j} c_{i,j} (p_i - p_j)^2 \Bigr),
\]
where \(A\) is large and the \(c_{i,j}\) are non-negative weights. The Gibbs sampler used is \textit{Glauber dynamics}: a single, uniformly chosen coordinate is updated in each step according to the conditional distribution given all the others.
The non-negative weights can be viewed as a weighted graph. Extending these weights to a symmetric matrix \(C = (c_{i,j})_{i,j=1}^d\) with diagonal \(c_{i,i} := \sum_{j \in [d] : j \ne i} c_{i,j}\), the weights generate a reversible, discrete-time Markov chain on \([d]\):
\[
\textstyle p_{i,j} = c_{i,j} / c_i, \quad\text{where}\quad c_i := \sum_{j \in [d] : j \ne i} c_{i,j}.
\]
It has equilibrium distribution \(c = (c_i)_{i=1}^d\).
Letting \(\Delta = I - P\) denote the Laplacian of the Markov chain,
\[
\textstyle \sum_{i,j \in [d] : i < j} c_{i,j} (p_i - p_j)^2 = \langle \mathbf p, \Delta \mathbf p \rangle, \quad\text{where}\quad \langle \mathbf x, \mathbf y \rangle := \sum_{i \in [d]} \pi_i x_i y_i.
\]
The main result analyses the mixing time of the Gibbs sampler, in the limit \(A \to \infty\) with other variables fixed. The mixing metric is the \(\ell_\infty\)-Wasserstein distance:
\[
\textstyle d_\infty(\mu, \pi) := \inf_{(X, Y)} \mathbb E( \| X - Y \|_\infty ),
\]
where the infimum is over all couplings \((X, Y)\) of \(\mu\) and \(\pi\). For \(\varepsilon \in (0,1)\), define
\[
\textstyle t_\mathsf{mix}(\varepsilon) := \inf\{ t \ge 0 \mid \sup_{\mathbf p \in [0,1]^d} d_\infty(K_t(\mathbf p), \pi_{A,C}) \le \varepsilon \},
\]
where \(K_t(\mathbf p)\) are the time-\(t\) transition probabilities of the Gibbs sampler started from \(\mathbf p \in [0,1]^d\) and \(\pi_{A,C}\) is the target distribution:
\[
\frac{d\pi_{A,C}}{d\mathbf p} \propto \exp\Bigl( - A^2 \sum_{i,j \in [d] : i < j} c_{i,j} (p_i - p_j)^2 \Bigr).
\]
In short, the mixing time is shown to be of order \(A^2\):
\[
t_\mathsf{mix}(\tfrac14 - \varepsilon) \gtrsim \tfrac \lambda\gamma A^2 \quad\text{and}\quad t_\mathsf{mix}(\varepsilon) \lesssim d A^2,
\]
where \(\lambda\) is the spectral gap of \(\Delta\) and \(\gamma\) the absolute spectral gap. The implicit constants in \(\gtrsim\)/\(\lesssim\) depend only on \(\varepsilon\).
This paper is motivated by work of de Finetti, who was particularly interested in understanding situations where \(\mathbf p\) is approximately concentrated around a main diagonal of the hypercube:
\[
\mathcal D := \{ p \mathbf 1 \mid p \in [0,1] \} \subseteq [0,1]^d, \quad\text{where}\quad \mathbf 1 := (1, \ldots, 1).
\]
This situation is referred to as \textit{almost exchangeability}, because it captures the idea that the \(d\) variables are almost indistinguishable.
Reviewer: Sam Olesker-Taylor (Coventry)Markov processes related to the stationary measure for the open KPZ equationhttps://zbmath.org/1528.600762024-03-13T18:33:02.981707Z"Bryc, Włodek"https://zbmath.org/authors/?q=ai:bryc.wlodzimierz"Kuznetsov, Alexey"https://zbmath.org/authors/?q=ai:kuznetsov.aleksey-mikhailovich|kuznetsov.alexey-s|kuznetsov.alexey"Wang, Yizao"https://zbmath.org/authors/?q=ai:wang.yizao"Wesołowski, Jacek"https://zbmath.org/authors/?q=ai:wesolowski.jacekSummary: We provide a probabilistic description of the stationary measures for the open KPZ on the spatial interval \([0, 1]\) in terms of a Markov process \(Y\), which is a Doob's \(h\) transform of the Brownian motion killed at an exponential rate. Our work builds on a recent formula of Corwin and Knizel which expresses the multipoint Laplace transform of the stationary solution of the open KPZ in terms of another Markov process \(\mathbb{T}\): the continuous dual Hahn process with Laplace variables taking on the role of time-points in the process. The core of our approach is to prove that the Laplace transforms of the finite dimensional distributions of \(Y\) and \(\mathbb{T}\) are equal when the time parameters of one process become the Laplace variables of the other process and vice versa.Space reduction for a class of multidimensional Markov chains: a summary and some applicationshttps://zbmath.org/1528.600772024-03-13T18:33:02.981707Z"He, Qi-Ming"https://zbmath.org/authors/?q=ai:he.qi-ming"Alfa, Attahiru Sule"https://zbmath.org/authors/?q=ai:alfa.attahiru-suleSummary: In this paper, we present examples of a class of Markov chains that occur frequently, but whose associated matrices are a challenge to construct efficiently. These are Markov chains that arise as a result of several identical Markov chains running in parallel. Specifically for the cases considered, both the infinitesimal generator matrix for the continuous case, and more so the transition probability matrix for the discrete equivalent, are complex to construct effectively and efficiently. We summarize the algorithms for constructing the associated matrices and present examples of applications, ranging from special queueing problems to reliability issues and order statistics. MATLAB subroutines are provided in an online supplement for the implementation of the algorithms.On Feller and strong Feller properties and irreducibility of regime-switching jump diffusion processes with countable regimeshttps://zbmath.org/1528.600782024-03-13T18:33:02.981707Z"Kunwai, Khwanchai"https://zbmath.org/authors/?q=ai:kunwai.khwanchai"Zhu, Chao"https://zbmath.org/authors/?q=ai:zhu.chaoThe Feller property and irreducibility are fundamental desirable properties of Markov processes. Motivated by control and optimization problems, this paper presents weak local non-Lipschitz conditions for Feller and strong Feller properties, as well as the irreducibility of regime-switching jump diffusion processes with countable regimes. The approach used for obtaining sufficient conditions for Feller and strong Feller properties is motivated by a gradient estimate for diffusion semigroups in the literature. While the technique is obtaining irreducibility under certain assumptions, it relies on an identity concerning the transition probability of the process. As an application, these conditions are used to obtain the existence of a unique invariant measure based on classical results of Markov processes.
Reviewer: Chuang Xu (Honolulu)A bidirectional hitting probability for the symmetric Hunt processeshttps://zbmath.org/1528.600792024-03-13T18:33:02.981707Z"Nishimori, Yasuhito"https://zbmath.org/authors/?q=ai:nishimori.yasuhitoSummary: We write \(\sigma_A\) the first hitting time of set \(A\) for the Hunt processes. Let \(B\) and \(B_R\) be compact sets, where \(B_R\) states far away from \(B\). We assume that the Hunt process is irreducible and conservative and satisfies the Feller property. We consider a relation of the hitting probability of \(B\) from \(B_R\) with the hitting probability of \(B_R\) from \(B\), without the spatial homogeneity. Our claim is that if the Hunt process satisfies the strong Feller property, then \(\lim_{x \to \infty} P_x ( \sigma_B \leq t ) = 0\) implies that \(\lim_{R \to \infty} P_y ( \sigma_{B_R} \leq t ) = 0\), for \(y \in B\). Additionally, if the Hunt process is \(m\)-symmetric, then both statements are equivalent.Transience of symmetric nonlocal Dirichlet formshttps://zbmath.org/1528.600802024-03-13T18:33:02.981707Z"Shiozawa, Yuichi"https://zbmath.org/authors/?q=ai:shiozawa.yuichiThe paper focuses on recurrence and transience of symmetric Markov processes generated by Dirichlet forms. It establishes transience criteria for a class of symmetric jump processes. Nonlocal Dirichlet forms have two kinds of the coefficients corresponding to the small and big jump parts. The results in this paper show that both of the coefficients do affect the sample path properties.
Reviewer: Oleg K. Zakusilo (Kyïv)Universal cutoff for Dyson Ornstein Uhlenbeck processhttps://zbmath.org/1528.600812024-03-13T18:33:02.981707Z"Boursier, Jeanne"https://zbmath.org/authors/?q=ai:boursier.jeanne"Chafaï, Djalil"https://zbmath.org/authors/?q=ai:chafai.djalil"Labbé, Cyril"https://zbmath.org/authors/?q=ai:labbe.cyrilSummary: We study the Dyson-Ornstein-Uhlenbeck diffusion process, an evolving gas of interacting particles. Its invariant law is the beta Hermite ensemble of random matrix theory, a non-product log-concave distribution. We explore the convergence to equilibrium of this process for various distances or divergences, including total variation, relative entropy, and transportation cost. When the number of particles is sent to infinity, we show that a cutoff phenomenon occurs: the distance to equilibrium vanishes abruptly at a critical time. A remarkable feature is that this critical time is independent of the parameter beta that controls the strength of the interaction, in particular the result is identical in the non-interacting case, which is nothing but the Ornstein-Uhlenbeck process. We also provide a complete analysis of the non-interacting case that reveals some new phenomena. Our work relies among other ingredients on convexity and functional inequalities, exact solvability, exact Gaussian formulas, coupling arguments, stochastic calculus, variational formulas and contraction properties. This work leads, beyond the specific process that we study, to questions on the high-dimensional analysis of heat kernels of curved diffusions.On a diffusion which stochastically restarts from moving random spatial positions: a non-renewal frameworkhttps://zbmath.org/1528.600822024-03-13T18:33:02.981707Z"da Silva, Telles Timóteo"https://zbmath.org/authors/?q=ai:da-silva.telles-timoteoSummary: We consider a diffusive particle that at random times, exponentially distributed with parameter \(\beta\), stops its motion and restarts from a moving random position \(Y(t)\) in space. The position \(X(t)\) of the particle and the restarts do not affect the dynamics of \(Y(t)\), so our framework constitutes in a non-renewal one. We exhibit the feasibility to build a rigorous general theory in this setup from the analysis of sample paths. To prove the stochastic process \(X(t)\) has a non-equilibrium steady-state, assumptions related to the confinement of \(Y(t)\) have to be imposed. In addition we design a detailed example where the random restart positions are provided by the paradigmatic Evans and Majumdar's diffusion with stochastic resettings [\textit{M. R. Evans} and \textit{S. N. Majumdar}, Phys. Rev. Lett. 106, No. 16, Article ID 160601, 4 p. (2011; \url{doi:10.1103/PhysRevLett.106.160601})], with resetting rate \(\beta_Y\). We show the ergodic property for the main process and for the stochastic process of jumps performed by the particle. A striking feature emerges from the examination of the jumps, since their negative covariance can be minimized with respect to both rates \(\beta\) and \(\beta_Y\), independently. Moreover we discuss the theoretical consequences that this non-renewal model entails for the analytical study of the mean first-passage time (FPT) and mean cost up to FPT.
{{\copyright} 2023 IOP Publishing Ltd}Ergodic properties of Fleming-Viot processes with selection.https://zbmath.org/1528.600832024-03-13T18:33:02.981707Z"Itatsu, Seiichi"https://zbmath.org/authors/?q=ai:itatsu.seiichi(no abstract)A note on the central limit theorem for the idleness process in a one-sided reflected Ornstein-Uhlenbeck modelhttps://zbmath.org/1528.600842024-03-13T18:33:02.981707Z"Mandjes, Michel"https://zbmath.org/authors/?q=ai:mandjes.michel"Spreij, Peter"https://zbmath.org/authors/?q=ai:spreij.peterSummary: In this short communication, we present a (functional) central limit theorem for the idleness process of a one-sided reflected Ornstein-Uhlenbeck proces.
{{\copyright} 2017 The Authors. Statistica Neerlandica {\copyright} 2017 VVS.}Off-diagonal heat kernel estimates for symmetric diffusions in a degenerate ergodic environmenthttps://zbmath.org/1528.600852024-03-13T18:33:02.981707Z"Taylor, Peter A."https://zbmath.org/authors/?q=ai:taylor.peter-aAuthor's abstract: We study a symmetric diffusion process on \(\mathbb{R}^d, d\geq 2\), in divergence form in a stationary and ergodic random environment. The coefficients are assumed to be degenerate and unbounded but satisfy a moment condition. We derive upper off-diagonal estimates on the heat kernel of this process for general speed measure. Lower off-diagonal estimates are also shown for a natural choice of speed measure, under an additional mixing assumption on the environment. Using these estimates, a scaling limit for the Green function is proven.
Reviewer: Ze-Chun Hu (Chengdu)On the Davis-Monroe problemhttps://zbmath.org/1528.600862024-03-13T18:33:02.981707Z"Yaskov, Pavel A."https://zbmath.org/authors/?q=ai:yaskov.pavel-aThe paper discusses the resolution of the Davis and Monroe problem concerning specific values of \(\varepsilon>0\) for which the distribution of the Brownian motion with nonlinear drift is equivalent or singular with respect to the Wiener measure on \(C[0, 1]\). It delves into the analysis of \(\mu_\varepsilon\), the distribution of \(B_\varepsilon = B + \varepsilon F\) as a random element of \(C[0, 1]\), where \(B\) represents the standard Brownian motion, \(F(t) = \sqrt{(t -\tau)^+}\), and \(\tau\) is a uniformly distributed random moment of time on \([0, 1]\), independent of \(B\). This problem is closely connected to the theory of Gaussian multiplicative chaos.
The authors highlight the criticality of power drift using the Cameron-Martin theorem, demonstrating that for \(F(t) \equiv t^\alpha\) and \(\alpha > 0\), the measures \(\mu_\varepsilon\) are either equivalent to \(\mu(\mu_\varepsilon\sim\mu)\) when \(\alpha > 1/2\) or singular with respect to \(\mu(\mu_\varepsilon\perp\mu)\) when \(\alpha\leq 1/2\). The document emphasizes the intricacy arising when \(\alpha = 1/2\) and the drift occurs at a random moment \(\tau\). It outlines findings from [\textit{B. Davis} and \textit{I. Monroe}, Ann. Probab. 12, 922--925 (1984; Zbl 0599.60041)], indicating that if \(F(t) \equiv \sqrt{(t-\tau)}^+\) and \(\tau\) possesses specific properties, \(\mu_\varepsilon\) is absolutely continuous with respect to \(\mu\) for \(\varepsilon < 2\) and singular for \(\varepsilon > \sqrt{8}\). The central contribution, presented as Theorem 1, investigates the unexplored territory for \(\varepsilon\in [2,\sqrt{8}]\).
Overall, the paper elucidates the connection between the nonlinear drift of Brownian motion, Gaussian multiplicative chaos theory, and the critical values of \(\varepsilon\) for which the measures \(\mu_\varepsilon\) align with or differ from the Wiener measure \(\mu\). It addresses the complexities introduced by a power drift occurring at a random moment, presenting a novel result for previously unexplored ranges of \(\varepsilon\).
Reviewer: Gerardo Hernandez-del-Valle (Ciudad de México)Poisson generalized Lindley process and its propertieshttps://zbmath.org/1528.600872024-03-13T18:33:02.981707Z"Cha, Ji Hwan"https://zbmath.org/authors/?q=ai:cha.ji-hwan"Badía, F. G."https://zbmath.org/authors/?q=ai:badia.francisco-germanSummary: In spite of the practical usefulness of the nonhomogeneous Poisson process, it still has some restrictions. To overcome these restrictions, the Poisson Lindley process has been recently developed and introduced in
[\textit{J. H. Cha}, Stat. Probab. Lett. 152, 74--81 (2019; Zbl 1451.60050)]. In this paper, we further generalize the Poisson Lindley process, so that the developed counting process model should have the restarting property and it should include the generalized Polya process as a special case. Some basic stochastic properties of the developed counting process model are derived. Dependence properties and stochastic comparisons are also discussed under a more general framework.A general class of shock models with dependent inter-arrival timeshttps://zbmath.org/1528.600882024-03-13T18:33:02.981707Z"Goyal, Dheeraj"https://zbmath.org/authors/?q=ai:goyal.dheeraj"Hazra, Nil Kamal"https://zbmath.org/authors/?q=ai:hazra.nil-kamal"Finkelstein, Maxim"https://zbmath.org/authors/?q=ai:finkelstein.maximSummary: We introduce and study a general class of shock models with dependent inter-arrival times of shocks that occur according to the homogeneous Poisson generalized gamma process. A lifetime of a system affected by a shock process from this class is represented by the convolution of inter-arrival times of shocks. This class contains many popular shock models, namely the extreme shock model, the generalized extreme shock model, the run shock model, the generalized run shock model, specific mixed shock models, etc. For systems operating under shocks, we derive and discuss the main reliability characteristics (namely the survival function, the failure rate function, the mean residual lifetime function and the mean lifetime) and study relevant stochastic comparisons. Finally, we provide some numerical examples and illustrate our findings by the application that considers an optimal mission duration policy.Failure rate-based models for systems subject to random shockshttps://zbmath.org/1528.600892024-03-13T18:33:02.981707Z"Wang, Guanjun"https://zbmath.org/authors/?q=ai:wang.guanjun"Liu, Peng"https://zbmath.org/authors/?q=ai:liu.peng.3"Shen, Lijuan"https://zbmath.org/authors/?q=ai:shen.lijuanSummary: Shock process that can increase the failure rate of the systems working under a stochastic environment is known as the shot-noise process. In the existing literature focusing on the shot-noise process, it is usually assumed that the failure rate increments caused by shocks are the same constants or identically distributed variables. However, such an assumption has limitations if the external environment is constantly changing or the shocks can weaken the system gradually. This article proposes an extended shock model in which the failure rate increments of systems caused by shocks need not be identical or identically distributed. The reliability functions of the systems as well as some reliability indices are derived explicitly. Some special cases are discussed. Besides, a preventive maintenance model is also investigated in which the system is replaced when the number of shocks reaches a fixed threshold. Finally, an example on a refrigeration system is provided to illustrate the proposed model.Probability density of the interval duration between events in the generalized MAP with its incomplete observabilityhttps://zbmath.org/1528.600902024-03-13T18:33:02.981707Z"Keba, Anastasia"https://zbmath.org/authors/?q=ai:keba.anastasia"Nezhel'skaya, Ludmila"https://zbmath.org/authors/?q=ai:nezhelskaya.ludmilaSummary: We consider a generalized MAP (Markovian Arrival Process) with an arbitrary number of states under conditions of its incomplete observability (in the presence of unextendable dead time of fixed duration). An explicit form of the probability density of the values of the interval duration between the moments of the events occurrence for a two-state flow is found.
For the entire collection see [Zbl 1517.68029].Loss probability in priority limited processing queueing systemhttps://zbmath.org/1528.600912024-03-13T18:33:02.981707Z"Yashina, Marina"https://zbmath.org/authors/?q=ai:yashina.marina-viktorovna"Tatashev, Alexander"https://zbmath.org/authors/?q=ai:tatashev.alexander-gennadjevich"de Alencar, Marcelo Sampaio"https://zbmath.org/authors/?q=ai:sampaio-de-alencar.marcelo(no abstract)Explosive growth for a constrained Hastings-Levitov aggregation modelhttps://zbmath.org/1528.600922024-03-13T18:33:02.981707Z"Berestycki, Nathanaël"https://zbmath.org/authors/?q=ai:berestycki.nathanael"Silvestri, Vittoria"https://zbmath.org/authors/?q=ai:silvestri.vittoriaSummary: We consider a constrained version of the \(\mathrm{HL}(0)\) Hastings-Levitov model of aggregation in the complex plane, in which particles can only attach to the part of the cluster that has already been grown. Although one might expect that this gives rise to a non-trivial limiting shape, we prove that the cluster grows explosively: in the upper half plane, the aggregate accumulates infinite diameter as soon as it reaches positive capacity. More precisely, we show that after \(nt\) particles of (half-plane) capacity \(1/(2n)\) have attached, the diameter of the shape is highly concentrated around \(\sqrt{t \log n} \), uniformly in \(t \in [0, T]\). This illustrates a new instability phenomenon for the growth of single trees/fjords in unconstrained \(\mathrm{HL}(0)\).Random interlacement is a factor of i.i.d.https://zbmath.org/1528.600932024-03-13T18:33:02.981707Z"Borbényi, Márton"https://zbmath.org/authors/?q=ai:borbenyi.marton"Ráth, Balázs"https://zbmath.org/authors/?q=ai:rath.balazs"Rokob, Sándor"https://zbmath.org/authors/?q=ai:rokob.sandorSummary: The random interlacement point process (introduced in [\textit{A.-S. Sznitman}, Ann. Math. (2) 171, No. 3, 2039--2087 (2010; Zbl 1202.60160)], generalized in [\textit{A. Teixeira}, Electron. J. Probab. 14, 1604--1627 (2009; Zbl 1192.60108)]) is a Poisson point process on the space of labeled doubly infinite nearest neighbour trajectories modulo time-shift on a transient graph \(G\). We show that the random interlacement point process on any transient transitive graph \(G\) is a factor of i.i.d., i.e., it can be constructed from a family of i.i.d. random variables indexed by vertices of the graph via an equivariant measurable map. Our proof uses a variant of the soft local time method (introduced in [\textit{S. Popov} and \textit{A. Teixeira}, J. Eur. Math. Soc. (JEMS) 17, No. 10, 2545--2593 (2015; Zbl 1329.60342)]) to construct the interlacement point process as the almost sure limit of a sequence of finite-length variants of the model with increasing length. We also discuss a more direct method of proving that the interlacement point process is a factor of i.i.d. which works if and only if \(G\) is non-unimodular.Last passage isometries for the directed landscapehttps://zbmath.org/1528.600942024-03-13T18:33:02.981707Z"Dauvergne, Duncan"https://zbmath.org/authors/?q=ai:dauvergne.duncanSummary: Consider the restriction of the directed landscape \(\mathcal{L}(x, s; y, t)\) to a set of the form \(\{x_1, \dots, x_k\} \times \{s_0\} \times \mathbb{R} \times \{t_0\}\). We show that on any such set, the directed landscape is given by a last passage problem across \(k\) locally Brownian functions. The \(k\) functions in this last passage isometry are built from certain marginals of the extended directed landscape. As applications of this construction, we show that the Airy difference profile is locally absolutely continuous with respect to Brownian local time, that the KPZ fixed point started from two narrow wedges has a Brownian-Bessel decomposition around its cusp point, and that the directed landscape is a function of its geodesic shapes.Optimal tail exponents in general last passage percolation via bootstrapping \& geodesic geometryhttps://zbmath.org/1528.600952024-03-13T18:33:02.981707Z"Ganguly, Shirshendu"https://zbmath.org/authors/?q=ai:ganguly.shirshendu"Hegde, Milind"https://zbmath.org/authors/?q=ai:hegde.milindSummary: We consider last passage percolation on \(\mathbb{Z}^2\) with general weight distributions, which is expected to be a member of the Kardar-Parisi-Zhang (KPZ) universality class. In this model, an oriented path between given endpoints which maximizes the sum of the i.i.d. weight variables associated to its vertices is called a geodesic. Under natural conditions of curvature of the limiting geodesic weight profile and stretched exponential decay of both tails of the point-to-point weight, we use geometric arguments to upgrade the tail assumptions to prove optimal upper and lower tail behavior with the exponents of 3/2 and 3 for the weight of the geodesic from \((1, 1)\) to \((r, r)\) for all large finite \(r\), and thus unearth a connection between the tail exponents and the characteristic KPZ weight fluctuation exponent of 1/3. The proofs merge several ideas which are not reliant on the exact form of the vertex weight distribution, including the well known super-additivity property of last passage values, concentration of measure behavior for sums of stretched exponential random variables, and geometric insights coming from the study of geodesics and more general objects called geodesic watermelons. Previous proofs of such optimal estimates have relied on hard analysis of precise formulas available only in integrable models. Our results illustrate a facet of universality in a class of KPZ stochastic growth models and provide a geometric explanation of the upper and lower tail exponents of the GUE Tracy-Widom distribution, the conjectured one point scaling limit of such models. The key arguments are based on an observation of general interest that super-additivity allows a natural iterative bootstrapping procedure to obtain improved tail estimates.Quasi-equilibria and click times for a variant of Muller's ratchethttps://zbmath.org/1528.600962024-03-13T18:33:02.981707Z"González Casanova, Adrián"https://zbmath.org/authors/?q=ai:casanova.adrian-gonzalez"Smadi, Charline"https://zbmath.org/authors/?q=ai:smadi.charline"Wakolbinger, Anton"https://zbmath.org/authors/?q=ai:wakolbinger.antonSummary: Consider a population of \(N\) individuals, each of them carrying a type in \(\mathbb{N}_0\). The population evolves according to a Moran dynamics with selection and mutation, where an individual of type \(k\) has the same selective advantage over all individuals with type \(k^\prime > k\), and type \(k\) mutates to type \(k + 1\) at a constant rate. This model is thus a variation of the classical Muller's ratchet: there the selective advantage is proportional to \(k^\prime - k\). For a regime of selection strength and mutation rates which is between the regimes of weak and strong selection/mutation, we obtain the asymptotic rate of the click times of the ratchet (i.e. the times at which the hitherto minimal (`best') type in the population is lost), and reveal the quasi-stationary type frequency profile between clicks. The large population limit of this profile is characterized as the normalized attractor of a ``dual'' hierarchical multitype logistic system, and also via the distribution of the final minimal displacement in a branching random walk with one-sided steps. An important role in the proofs is played by a graphical representation of the model, both forward and backward in time, and a central tool is the ancestral selection graph decorated by mutations.Percolation in the Boolean model with convex grains in high dimensionhttps://zbmath.org/1528.600972024-03-13T18:33:02.981707Z"Gouéré, Jean-Baptiste"https://zbmath.org/authors/?q=ai:gouere.jean-baptiste"Labéy, Florestan"https://zbmath.org/authors/?q=ai:labey.florestanSummary: We investigate percolation in the Boolean model with convex grains in high dimension. For each dimension \(d\), one fixes a compact, convex and symmetric set \(K \subset \mathbb{R}^d\) with non empty interior. In a first setting, the Boolean model is a reunion of translates of \(K\). In a second setting, the Boolean model is a reunion of translates of \(K\) or \({\rho} K\) for a further parameter \({\rho} \in(1, 2)\). We give the asymptotic behavior of the percolation probability and of the percolation threshold in the two settings.Selected problems in probability theoryhttps://zbmath.org/1528.600982024-03-13T18:33:02.981707Z"Grimmett, Geoffrey R."https://zbmath.org/authors/?q=ai:grimmett.geoffrey-rSummary: This celebratory article contains a personal and idiosyncratic selection of a few open problems in discrete probability theory. These include certain well-known questions concerning Lorentz scatterers and self-avoiding walks, and also some problems of percolation-type. The author hopes the reader will find something to leaven winter evenings, and perhaps even a project for the longer term.
For the entire collection see [Zbl 1515.01005].Slightly supercritical percolation on non-amenable graphs. I: The distribution of finite clustershttps://zbmath.org/1528.600992024-03-13T18:33:02.981707Z"Hutchcroft, Tom"https://zbmath.org/authors/?q=ai:hutchcroft.tomSummary: We study the distribution of finite clusters in slightly supercritical \((p \downarrow p_c)\) Bernoulli bond percolation on transitive non-amenable graphs, proving in particular that if \(G\) is a transitive non-amenable graph satisfying the \(L^2\) \textit{boundedness condition} \((p_c<p_{2\rightarrow 2})\) and \(K\) denotes the cluster of the origin and then there exists \(\delta >0\) such that if \(p\in (p_c-\delta ,p_c+\delta)\), then
\[
\begin{aligned}
\mathbf{P}_p(n \leqslant |K| < \infty) &\asymp n^{-1/2} \,{\exp} {\left[ -\Theta{\left(|p-p_c|^2 n\right)} \right]}
\end{aligned}
\]
and
\[
\begin{aligned}
\mathbf{P}_p(r \leqslant \operatorname{Rad}(K) < \infty) &\asymp r^{-1} \exp{[ -\Theta (|p-p_c| r) ]}
\end{aligned}
\]
for every \(n\), \(r\geqslant 1\), where all implicit constants depend only on \(G\). We deduce in particular that the critical exponents \(\gamma^{\prime}\) and \(\Delta^{\prime}\) describing the rate of growth of the moments of a finite cluster as \(p \downarrow p_c\) take their mean-field values of 1 and 2, respectively. These results apply in particular to Cayley graphs of non-elementary hyperbolic groups, to products with trees, and to transitive graphs of spectral radius \(\rho <1/2\). In particular, every finitely generated non-amenable group has a Cayley graph to which these results apply. They are new for graphs that are not trees. The corresponding facts are yet to be understood on \(\mathbb{Z}^d\) even for \(d\) very large. In a second paper in this series, we will apply these results to study the geometric and spectral properties of infinite slightly supercritical clusters in the same setting.
{{\copyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}Continuum percolation in a nonstabilizing environmenthttps://zbmath.org/1528.601002024-03-13T18:33:02.981707Z"Jahnel, Benedikt"https://zbmath.org/authors/?q=ai:jahnel.benedikt"Jhawar, Sanjoy Kumar"https://zbmath.org/authors/?q=ai:jhawar.sanjoy-kumar"Anh Duc Vu"https://zbmath.org/authors/?q=ai:anh-duc-vu.Summary: We prove phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson-Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox-Boolean model. The phase transitions are established under individually as well as jointly varying parameters. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in
[\textit{C. Hoffman}, Commun. Math. Phys. 254, No. 1, 1--22 (2005; Zbl 1128.60084)].Classification of stationary distributions for the stochastic vertex modelshttps://zbmath.org/1528.601012024-03-13T18:33:02.981707Z"Lin, Yier"https://zbmath.org/authors/?q=ai:lin.yierSummary: In this paper, we study the stationary distributions for the stochastic vertex models. Our main focus is the stochastic six vertex (S6V) model. We show that the extremal stationary distributions of the S6V model are given by product Bernoulli measures. Moreover, for the S6V model under a moving frame of speed 1, we show that the extremal stationary distributions are given by product Bernoulli measures and blocking measures. Finally, we generalize our results to the stochastic higher spin six vertex model. Our proof relies on the coupling of the S6V models introduced in [\textit{A. Aggarwal}, Commun. Math. Phys. 376, No. 1, 681--746 (2020; Zbl 1445.60070)], the analysis of current and the method of fusion.Balancing conservative and disruptive growth in the voter modelhttps://zbmath.org/1528.601022024-03-13T18:33:02.981707Z"Ross, Robert J. H."https://zbmath.org/authors/?q=ai:ross.robert-j-h"Fontana, Walter"https://zbmath.org/authors/?q=ai:fontana.walterSummary: We are concerned with how the implementation of growth determines the expected number of state-changes in a growing self-organizing process. With this problem in mind, we examine two versions of the voter model on a one-dimensional growing lattice. Our main result asserts that the expected number of state-changes before an absorbing state is found can be controlled by balancing the conservative and disruptive forces of growth. This is because conservative growth preserves the self-organization of the voter model as it searches for an absorbing state, whereas disruptive growth undermines this self-organization. In particular, we focus on controlling the expected number of state-changes as the rate of growth tends to zero or infinity in the limit. These results illustrate how growth can affect the costs of self-organization and so are pertinent to the physics of growing active matter.Contact process on a dynamical long range percolationhttps://zbmath.org/1528.601032024-03-13T18:33:02.981707Z"Seiler, Marco"https://zbmath.org/authors/?q=ai:seiler.marco"Sturm, Anja"https://zbmath.org/authors/?q=ai:sturm.anja-kSummary: In this paper we introduce a contact process on a dynamical long range percolation (CPDLP) defined on a complete graph \((V, \mathcal{E})\). A dynamical long range percolation is a Feller process defined on the edge set \(\mathcal{E}\), which assigns to each edge the state of being open or closed independently. The state of an edge \(e\) is updated at rate \(v_e\) and is open after the update with probability \(p_e\) and closed otherwise. The contact process is then defined on top of this evolving random environment using only open edges for infection while recovery is independent of the background. First, we conclude that an upper invariant law exists and that the phase transitions of survival and non-triviality of the upper invariant coincide. We then formulate a comparison with a contact process with a specific infection kernel which acts as a lower bound. Thus, we obtain an upper bound for the critical infection rate. We also show that if the probability that an edge is open is low for all edges then the CPDLP enters an immunization phase, i.e. it will not survive regardless of the value of the infection rate. Furthermore, we show that on \(V = \mathbb{Z}\) and under suitable conditions on the rates of the dynamical long range percolation the CPDLP will almost surely die out if the update speed converges to zero for any given infection rate \(\lambda\).Contact process in an evolving random environmenthttps://zbmath.org/1528.601042024-03-13T18:33:02.981707Z"Seiler, Marco"https://zbmath.org/authors/?q=ai:seiler.marco"Sturm, Anja"https://zbmath.org/authors/?q=ai:sturm.anja-kSummary: In this paper we introduce a contact process in an evolving random environment (CPERE) on a connected and transitive graph with bounded degree, where we assume that this environment is described through an ergodic spin systems with finite range. We show that under a certain growth condition the phase transition of survival is independent of the initial configuration of the process. We study the invariant laws of the CPERE and show that under aforementioned growth condition the phase transition for survival coincides with the phase transition of non-triviality of the upper invariant law. Furthermore, we prove continuity properties for the survival probability and derive equivalent conditions for complete convergence, in an analogous way as for the classical contact process. We then focus on the special case, where the evolving random environment is described through a dynamical percolation. We show that the contact process on a dynamical percolation on the \(d\)-dimensional integers dies out almost surely at criticality and complete convergence holds for all parameter choices. In the end we derive some comparison results between a dynamical percolation and ergodic spin systems with finite range such that we get bounds on the survival probability of a contact process in an evolving random environment and we determine in this case that complete convergence holds in a certain parameter regime.On the cost of the bubble set for random interlacementshttps://zbmath.org/1528.601052024-03-13T18:33:02.981707Z"Sznitman, Alain-Sol"https://zbmath.org/authors/?q=ai:sznitman.alain-solThe paper treats random interlacements of \(\mathbb{Z}^d\) (for \(d\geq 3\)) in the strongly percolative regime. Informally, the random interlacement at level \(u>0\) is a random subset \(\mathcal {I}^u\) of \(\mathbb{Z}^d\) that describes the picture left by the trace of a random walk on a large discrete torus, run up to a number of steps that is proportional to the size of the torus.
As \(u\) increases, the complement \(\mathcal V^u\) of \(\mathcal {I}^u\) in \(\mathbb{Z}^d\) thins out, and there is a value \(u_\ast>0\), called the \emph{critical level for percolation} such that \(\mathcal V^u\) has an infinite component when \(u< u_\ast\) and all connected components of \(\mathcal V^u\) are finite when \(u>u_\ast\). The assumption of \emph{strong criticality} made in the present paper is stronger: informally strongly percolative means local presence and local uniqueness of an infinite cluster of \(\mathcal{V}^u\).
There is another threshold \(\overline u\leq u_\ast\), called the critical level for strong percolation, such that strong criticality holds whenever \(u<\overline u\). Thus, the paper assumes that \(u<\overline u\). It is conjectured, but not proved, that \(u_\ast = \overline u\).
The present article studies the occurrence of the fraction of sites in a discrete box that get disconnected by the interlacements from the boundary of a box twice the size. An upper bound for the exponential decay of the probability of such sites exceeding a given threshold is the main result. This continues previous work by the author on this subject. In particular, if the conjecture that \(u_\ast=\overline u\) turns out to be true (as expected), the upper bound derived in this paper matches to first order a previously known lower bound.
More formally, let \(D_N=[-N,N]^d\cap \mathbb{Z}\) and \(S_N\) be the set of \(x\in\mathbb{Z}^d\) with sup-norm equal to \(N\). For \(r>0\) define \(\mathcal{C}_r^u\) to be the connected component of \(S_r\) in \(\mathcal {V}^u\cup S_r\), so that the set of points in \(D_N=[-N,N]^d\cap \mathbb{Z}\) that are disconnected from \(S_{2N}\)by \(\mathcal {I}^u\) may be written \(D_N\setminus C_{2N}^u\).
The main result of the paper is a large deviations upper bound of speed \(N^{d-2}\) for the probability of the event \(\mathcal {A}_N\) that \(D_N\setminus C_{2N}^u\) contains more than \(v |D_N|\) elements, for a suitably chosen parameter \(v\): \[ \limsup_{N\to\infty} \frac1{N^{d-2}} \log \mathbb{P}(A_N) \leq -\widehat{J}_{u,v}, \] where the function \(\widehat{J}_{u,v}\) is given by the minimum of \(\frac1{2d} \int_{\mathbb{R}^d} |\nabla \phi(z)|^2 dz\) over suitable positive, locally integrable functions \(\phi\) on \(\mathbb{R}^d\) with finite Dirichlet energy.
The results significantly improve the findings in the author's previous work (see [Probab. Theory Relat. Fields 167, No. 1--2, 1--44 (2017; Zbl 1365.60080); Probab. Math. Phys. 2, No. 3, 563--611 (2021; Zbl 1483.60149)]). In particular, they match up with a previously known large deviations lower bound with the same speed and function \(\widehat{J}_{u,v}\) to give a full large deviations principle if the conjecture \(\overline u=u_\ast\) holds.
Reviewer: Janosch Ortmann (Montréal)Moment characterization of the weak disorder phase for directed polymers in a class of unbounded environmentshttps://zbmath.org/1528.601062024-03-13T18:33:02.981707Z"Fukushima, Ryoki"https://zbmath.org/authors/?q=ai:fukushima.ryoki"Junk, Stefan"https://zbmath.org/authors/?q=ai:junk.stefanSummary: For a directed polymer model in random environment, a characterization of the weak disorder phase in terms of the moment of the renormalized partition function has been proved in [the second author, Commun. Math. Phys. 389, No. 2, 1087--1097 (2022; Zbl 1489.82101)]. We extend this characterization to a large class of unbounded environments which includes many commonly used distributions.Almost sure behavior for the local time of a diffusion in a spectrally negative Lévy environmenthttps://zbmath.org/1528.601072024-03-13T18:33:02.981707Z"Véchambre, Grégoire"https://zbmath.org/authors/?q=ai:vechambre.gregoireIn this paper, the author studies the almost sure asymptotic behavior of the supremum of the local time for a transient sub-ballistic diffusion in a spectrally negative Lévy environment. The proper renormalizations for the extremely large values of the supremum of the local time are given. As a corollary of these results, it is noted that the renormalization of the extremely large values of the supremum of the local time is determined by the asymptotic behavior of the Laplace exponent of the Lévy environment. Moreover, this renormalization is greater than the known renormalization for the recurrent case.
Reviewer: Utkir A. Rozikov (Tashkent)On the size of earthworm's trailhttps://zbmath.org/1528.601082024-03-13T18:33:02.981707Z"Burdzy, Krzysztof"https://zbmath.org/authors/?q=ai:burdzy.krzysztof"Feng, Shi"https://zbmath.org/authors/?q=ai:feng.shi"Shiraishi, Daisuke"https://zbmath.org/authors/?q=ai:shiraishi.daisukeSummary: We investigate the number of holes created by an ``earthworm'' moving on the two-dimensional integer lattice. The earthworm is modeled by a simple random walk. At the initial time, all vertices are filled with grains of soil except for the position of the earthworm. At each step, the earthworm pushes the soil in the direction of its motion. It leaves a hole (an empty vertex with no grain of soil) behind it. If there are holes in front of the earthworm (in the direction of its step), the closest hole is filled with a grain of soil. Thus the number of holes increases by 1 or remains unchanged at every step. We show that the number of holes is at least \(\mathcal{O}(n^{3 \slash 4})\) after \(n\) steps.On the model of random walk with multiple memory structurehttps://zbmath.org/1528.601092024-03-13T18:33:02.981707Z"Arkashov, N. S."https://zbmath.org/authors/?q=ai:arkashov.n-sSummary: A model of one-dimensional random walk based on the memory flow phenomenology is constructed. In this model, the jumps of the random walk process have a convolution structure formed on the basis of a finite sequence of memory functions and a stationary, generally speaking, non-Gaussian sequence. A physical interpretation of memory functions and the stationary sequence is given. A limit theorem in the metric space \(D[0, 1]\) for the normalized walk process is obtained.Non-geometric rough paths on manifoldshttps://zbmath.org/1528.601102024-03-13T18:33:02.981707Z"Armstrong, John"https://zbmath.org/authors/?q=ai:armstrong.john"Brigo, Damiano"https://zbmath.org/authors/?q=ai:brigo.damiano"Cass, Thomas"https://zbmath.org/authors/?q=ai:cass.thomas-r"Rossi Ferrucci, Emilio"https://zbmath.org/authors/?q=ai:ferrucci.emilio-rossiThe authors present a theory of manifold-valued rough paths of bounded \(p\)-variation for \(p<3\) (which are not required to be geometric) and their controlled integrands in a coordinate-free manner by using pushforwards and pullbacks through charts (showing how the choice of a linear connection gives rise to a definition of rough integral), and, consequently, they define rough differential equations in a similar spirit. Next they extend the theory of constrained rough paths to non-geometric integrators and controlled integrands more general than \(1\)-forms in the extrinsic Riemannian framework, and finally, they construct a parallel transport along rough paths and the resulting notion of Cartan development.
Reviewer: Martin Ondreját (Praha)A combinatorial approach to geometric rough paths and their controlled pathshttps://zbmath.org/1528.601112024-03-13T18:33:02.981707Z"Cass, Thomas"https://zbmath.org/authors/?q=ai:cass.thomas-r"Driver, Bruce K."https://zbmath.org/authors/?q=ai:driver.bruce-k"Litterer, Christian"https://zbmath.org/authors/?q=ai:litterer.christian"Rossi Ferrucci, Emilio"https://zbmath.org/authors/?q=ai:ferrucci.emilio-rossiSummary: We develop the structure theory for transformations of weakly geometric rough paths of bounded \(1 < p\)-variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. We derive an explicit combinatorial expression for the rough path lift of a controlled path, and use it to obtain fundamental identities such as the associativity of the rough integral, the adjunction between pushforwards and pullbacks and a change of variables formula for rough differential equations (RDEs). As applications we define rough paths, rough integration and RDEs on manifolds, extending the results of \textit{T. Cass} et al. [Proc. Lond. Math. Soc. (3) 111, No. 6, 1471--1518 (2015; Zbl 1338.60137)] to the case of arbitrary \(p\).Center manifolds for rough partial differential equationshttps://zbmath.org/1528.601122024-03-13T18:33:02.981707Z"Kuehn, Christian"https://zbmath.org/authors/?q=ai:kuhn.christian"Neamţu, Alexandra"https://zbmath.org/authors/?q=ai:neamtu.alexandraIn this paper, the authors deal with the existence of center manifolds for stochastic partial differential equations with nonlinear drift driven by \(\gamma\)-Holder rough paths with \(\gamma \in ({\frac 13}, {\frac 12})\). The main result is proved by the Lyapunov-Perron method along with the rough paths theory and the semigroup theory. As an application, the authors show the existence of center manifolds for a class of reaction-diffusion equations and the Swift-Hohenberg equations.
Reviewer: Bixiang Wang (Socorro)Moment-based estimation for parameters of general inverse subordinatorhttps://zbmath.org/1528.620142024-03-13T18:33:02.981707Z"Grzesiek, Aleksandra"https://zbmath.org/authors/?q=ai:grzesiek.aleksandra"Połoczański, Rafał"https://zbmath.org/authors/?q=ai:poloczanski.rafal"Kumar, Arun"https://zbmath.org/authors/?q=ai:kumar.arun.1"Wyłomańska, Agnieszka"https://zbmath.org/authors/?q=ai:wylomanska.agnieszkaSummary: In recent years the processes with anomalous diffusive dynamics have been widely discussed in the literature. The classic example of the anomalous diffusive models is the continuous time random walk (CTRW) which is a natural generalization of the random walk model. One of the fundamental properties of the classical CTRW is the fact that in the limit it tends to the Brownian motion subordinated by the so-called \(\beta\)-stable subordinator when the mean of waiting times is infinite. One can consider the generalization of such subordinated model by taking general inverse subordinator instead of the \(\beta\)-stable one as a time-change. The inverse subordinator is the first exit time of the non-decreasing Lévy process also called subordinator. In this paper we consider the Brownian motion delayed by general inverse subordinator. The main attention is paid to the estimation method of the parameters of the general inverse subordinator in the considered model. We propose a novel estimation technique based on the discretization of the subordinator's distribution. Using this approach we demonstrate that the distribution of the constant time periods, visible in the trajectory of the considered model, can be described by the so-called modified cumulative distribution function. This paper is an extension of the authors' previous article where a similar approach was applied, however here we focus on moment-based estimation and compare it with other popular methods of estimation. The effectiveness of the new algorithm is verified using the Monte Carlo approach.Some monotonicity results for stochastic kriging metamodels in sequential settingshttps://zbmath.org/1528.620412024-03-13T18:33:02.981707Z"Wang, Bing"https://zbmath.org/authors/?q=ai:wang.bing.3|wang.bing.1"Hu, Jiaqiao"https://zbmath.org/authors/?q=ai:hu.jiaqiaoSummary: Stochastic Kriging (SK) and stochastic kriging with gradient estimators (SKG) are useful methods for effectively approximating the response surface of a simulation model. In this paper, we show that in a fully sequential setting when all model parameters are known, the mean squared errors of the optimal SK and SKG predictors are monotonically decreasing as the number of design points increases. In addition, we prove, under appropriate conditions, that the use of gradient information in the SKG framework generally improves the prediction performance of SK. Motivated by these findings, we propose a sequential procedure for adaptively choosing design points and simulation replications in obtaining SK (SKG) predictors with desired levels of fidelity. We justify the validity of the procedure and carry out numerical experiments to illustrate its performance.Poissonian two-armed bandit: a new approachhttps://zbmath.org/1528.620542024-03-13T18:33:02.981707Z"Kolnogorov, A. V."https://zbmath.org/authors/?q=ai:kolnogorov.alexander-vSummary: We consider a new approach to the continuous-time two-armed bandit problem in which incomes are described by Poisson processes. For this purpose, first, the control horizon is divided into equal consecutive half-intervals in which the strategy remains constant, and the incomes arrive in batches corresponding to these half-intervals. For finding the optimal piecewise constant Bayesian strategy and its corresponding Bayesian risk, a recursive difference equation is derived. The existence of a limiting value of the Bayesian risk when the number of half-intervals grows infinitely is established, and a partial differential equation for finding it is derived. Second, unlike previously considered settings of this problem, we analyze the strategy as a function of the current history of the controlled process rather than of the evolution of the posterior distribution. This removes the requirement of finiteness of the set of admissible parameters, which was imposed in previous settings. Simulation shows that in order to find the Bayesian and minimax strategies and risks in practice, it is sufficient to partition the arriving incomes into 30 batches. In the case of the minimax setting, it is shown that optimal processing of arriving incomes one by one is not more efficient than optimal batch processing if the control horizon grows infinitely.On the convergence of two types of estimators of quadratic variationhttps://zbmath.org/1528.620552024-03-13T18:33:02.981707Z"Yu, Xisheng"https://zbmath.org/authors/?q=ai:yu.xisheng(no abstract)Advanced multilevel Monte Carlo methodshttps://zbmath.org/1528.650042024-03-13T18:33:02.981707Z"Jasra, Ajay"https://zbmath.org/authors/?q=ai:jasra.ajay"Law, Kody"https://zbmath.org/authors/?q=ai:law.kody-j-h"Suciu, Carina"https://zbmath.org/authors/?q=ai:suciu.carinaSummary: This article reviews the application of some advanced Monte Carlo techniques in the context of multilevel Monte Carlo (MLMC). MLMC is a strategy employed to compute expectations, which can be biassed in some sense, for instance, by using the discretization of an associated probability law. The MLMC approach works with a hierarchy of biassed approximations, which become progressively more accurate and more expensive. Using a telescoping representation of the most accurate approximation, the method is able to reduce the computational cost for a given level of error versus i.i.d. sampling from this latter approximation. All of these ideas originated for cases where exact sampling from couples in the hierarchy is possible. This article considers the case where such exact sampling is not currently possible. We consider some Markov chain Monte Carlo and sequential Monte Carlo methods, which have been introduced in the literature, and we describe different strategies that facilitate the application of MLMC within these methods.
{{\copyright} 2020 The Authors. International Statistical Review {\copyright} 2020 International Statistical Institute}Stochastic modified equations for the asynchronous stochastic gradient descenthttps://zbmath.org/1528.650052024-03-13T18:33:02.981707Z"An, Jing"https://zbmath.org/authors/?q=ai:an.jing"Lu, Jianfeng"https://zbmath.org/authors/?q=ai:lu.jianfeng"Ying, Lexing"https://zbmath.org/authors/?q=ai:ying.lexingSummary: We propose stochastic modified equations (SMEs) for modelling the asynchronous stochastic gradient descent (ASGD) algorithms. The resulting SME of Langevin type extracts more information about the ASGD dynamics and elucidates the relationship between different types of stochastic gradient algorithms. We show the convergence of ASGD to the SME in the continuous time limit, as well as the SME's precise prediction to the trajectories of ASGD with various forcing terms. As an application, we propose an optimal mini-batching strategy for ASGD via solving the optimal control problem of the associated SME.Strong convergence of semi-implicit split-step methods for SDE with locally Lipschitz coefficientshttps://zbmath.org/1528.650062024-03-13T18:33:02.981707Z"İzgi, Burhaneddin"https://zbmath.org/authors/?q=ai:izgi.burhaneddin"Çetin, Coşkun"https://zbmath.org/authors/?q=ai:cetin.coskunSummary: We discuss mean-square strong convergence properties for numerical solutions of a class of stochastic differential equations with super-linear drift terms using semi-implicit split-step methods. Under a one-sided Lipschitz condition on the drift term and a global Lipschitz condition on the diffusion term, we show that these numerical procedures yield the usual strong convergence rate of 1/2. We also present simulation-based applications including stochastic logistic growth equations, and compare their empirical convergence with some alternate methods.Computing committors in collective variables via Mahalanobis diffusion mapshttps://zbmath.org/1528.650072024-03-13T18:33:02.981707Z"Evans, Luke"https://zbmath.org/authors/?q=ai:evans.luke"Cameron, Maria K."https://zbmath.org/authors/?q=ai:cameron.maria-kourkina"Tiwary, Pratyush"https://zbmath.org/authors/?q=ai:tiwary.pratyushSummary: The study of rare events in molecular and atomic systems such as conformal changes and cluster rearrangements has been one of the most important research themes in chemical physics. Key challenges are associated with long waiting times rendering molecular simulations inefficient, high dimensionality impeding the use of PDE-based approaches, and the complexity or breadth of transition processes limiting the predictive power of asymptotic methods. Diffusion maps are promising algorithms to avoid or mitigate all these issues. We adapt the diffusion map with Mahalanobis kernel proposed by \textit{A. Singer} and \textit{R. R. Coifman} [Appl. Comput. Harmon. Anal. 25, No. 2, 226--239 (2008; Zbl 1144.62044)] for the SDE describing molecular dynamics in collective variables in which the diffusion matrix is position-dependent and, unlike the case considered by Singer and Coifman, is not associated with a diffeomorphism. We offer an elementary proof showing that one can approximate the generator for this SDE discretized to a point cloud via the Mahalanobis diffusion map. We use it to calculate the committor functions in collective variables for two benchmark systems: alanine dipeptide, and Lennard-Jones-7 in 2D. For validating our committor results, we compare our committor functions to the finite-difference solution or by conducting a ``committor analysis'' as used by molecular dynamics practitioners. We contrast the outputs of the Mahalanobis diffusion map with those of the standard diffusion map with isotropic kernel and show that the former gives significantly more accurate estimates for the committors than the latter.Higher order time discretization method for the stochastic Stokes equations with multiplicative noisehttps://zbmath.org/1528.650842024-03-13T18:33:02.981707Z"Vo, Liet"https://zbmath.org/authors/?q=ai:vo.lietSummary: In this paper, we propose a new approach for the time-discretization of the incompressible stochastic Stokes equations with multiplicative noise. Our new strategy is based on the classical Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly 1 for both velocity and pressure approximations. The proof is based on a new Hölder continuity estimate of the velocity solution. While the errors of the velocity approximation are estimated in the standard \(L^2\)- and \(H^1\)-norms, the pressure errors are carefully analyzed in a special norm because of the low regularity of the pressure solution. In addition, a new interpretation of the pressure solution, which is very useful in computation, is also introduced. Numerical experiments are also provided to validate the error estimates and their sharpness.Nonlocal PageRankhttps://zbmath.org/1528.680422024-03-13T18:33:02.981707Z"Cipolla, Stefano"https://zbmath.org/authors/?q=ai:cipolla.stefano"Durastante, Fabio"https://zbmath.org/authors/?q=ai:durastante.fabio"Tudisco, Francesco"https://zbmath.org/authors/?q=ai:tudisco.francescoSummary: In this work we introduce and study a nonlocal version of the PageRank. In our approach, the random walker explores the graph using longer excursions than just moving between neighboring nodes. As a result, the corresponding ranking of the nodes, which takes into account a \textit{long-range interaction} between them, does not exhibit concentration phenomena typical of spectral rankings which take into account just local interactions. We show that the predictive value of the rankings obtained using our proposals is considerably improved on different real world problems.Probabilistic total store orderinghttps://zbmath.org/1528.680742024-03-13T18:33:02.981707Z"Abdulla, Parosh Aziz"https://zbmath.org/authors/?q=ai:abdulla.parosh-aziz"Atig, Mohamed Faouzi"https://zbmath.org/authors/?q=ai:atig.mohamed-faouzi"Agarwal, Raj Aryan"https://zbmath.org/authors/?q=ai:agarwal.raj-aryan"Godbole, Adwait"https://zbmath.org/authors/?q=ai:godbole.adwait-amit"S., Krishna"https://zbmath.org/authors/?q=ai:s.krishnaSummary: We present \textit{Probabilistic Total Store Ordering (PTSO)} -- a probabilistic extension of the classical TSO semantics. For a given (finite-state) program, the operational semantics of PTSO induces an infinite-state Markov chain. We resolve the inherent non-determinism due to process schedulings and memory updates according to given probability distributions. We provide a comprehensive set of results showing the decidability of several properties for PTSO, namely (i) \textit{Almost-Sure (Repeated) Reachability}: whether a run, starting from a given initial configuration, almost surely visits (resp. almost surely repeatedly visits) a given set of target configurations. (ii) \textit{Almost-Never (Repeated) Reachability}: whether a run from the initial configuration, almost never visits (resp. almost never repeatedly visits) the target. (iii) \textit{Approximate Quantitative (Repeated) Reachability}: to approximate, up to an arbitrary degree of precision, the measure of runs that start from the initial configuration and (repeatedly) visit the target. (iv) \textit{Expected Average Cost}: to approximate, up to an arbitrary degree of precision, the expected average cost of a run from the initial configuration to the target. We derive our results through a nontrivial combination of results from the classical theory of (infinite-state) Markov chains, the theories of \textit{decisive} and \textit{eager} Markov chains, specific techniques from combinatorics, as well as, decidability and complexity results for the classical (non-probabilistic) TSO semantics. As far as we know, this is the first work that considers probabilistic verification of programs running on weak memory models.
For the entire collection see [Zbl 1516.68024].Variance reduction in stochastic reaction networks using control variateshttps://zbmath.org/1528.681082024-03-13T18:33:02.981707Z"Backenköhler, Michael"https://zbmath.org/authors/?q=ai:backenkohler.michael"Bortolussi, Luca"https://zbmath.org/authors/?q=ai:bortolussi.luca"Wolf, Verena"https://zbmath.org/authors/?q=ai:wolf.verenaSummary: Monte Carlo estimation in plays a crucial role in stochastic reaction networks. However, reducing the statistical uncertainty of the corresponding estimators requires sampling a large number of trajectories. We propose control variates based on the statistical moments of the process to reduce the estimators' variances. We develop an algorithm that selects an efficient subset of infinitely many control variates. To this end, the algorithm uses resampling and a redundancy-aware greedy selection. We demonstrate the efficiency of our approach in several case studies.
For the entire collection see [Zbl 1516.68022].Conclusive discrimination by \(N\) sequential receivers between \(r\geqslant 2\) arbitrary quantum stateshttps://zbmath.org/1528.810332024-03-13T18:33:02.981707Z"Loubenets, E. R."https://zbmath.org/authors/?q=ai:loubenets.elena-r"Namkung, M."https://zbmath.org/authors/?q=ai:namkung.minSummary: In the present paper, we develop a general mathematical framework for discrimination between \(r\geqslant 2\) quantum states by \(N\geqslant 1\) sequential receivers for the case in which every receiver obtains a conclusive result. This type of discrimination constitutes an \(N\)-sequential extension of the minimum-error discrimination by one receiver. The developed general framework, which is valid for a conclusive discrimination between any number \(r\geqslant 2\) of quantum states, pure or mixed, of an arbitrary dimension and any number \(N\geqslant 1\) of sequential receivers, is based on the notion of a quantum state instrument, and this allows us to derive new important general results. In particular, we find a general condition on \(r\geqslant 2\) quantum states under which, within the strategy in which all types of receivers' quantum measurements are allowed, the optimal success probability of the \(N\)-sequential conclusive discrimination between these \(r\geqslant 2\) states is equal to that of the first receiver for any number \(N\geqslant 2\) of further sequential receivers and specify the corresponding optimal protocol. Furthermore, we extend our general framework to include an \(N\)-sequential conclusive discrimination between \(r\geqslant 2\) arbitrary quantum states under a noisy communication. As an example, we analyze analytically and numerically a two-sequential conclusive discrimination between two qubit states via depolarizing quantum channels. The derived new general results are important both from the theoretical point of view and for the development of a successful multipartite quantum communication via noisy quantum channels.Mentor initiated controlled bi-directional remote state preparation scheme for \((2 \iff 4)\)-qubit entangled states in noisy channelhttps://zbmath.org/1528.810362024-03-13T18:33:02.981707Z"Choudhury, Binayak S."https://zbmath.org/authors/?q=ai:choudhury.binayak-samadder"Mandal, Manoj Kumar"https://zbmath.org/authors/?q=ai:mandal.manoj-kumar"Samanta, Soumen"https://zbmath.org/authors/?q=ai:samanta.soumenSummary: In this paper we present a bi-directional protocol for mutual remote preparation of a two and a four-qubit non-maximally entangled state where the parties intending to remotely prepare the respective states are not initially entangled. There is a controller of the protocol who oversees the performances of other parties and acts to signal for the execution of the final step in the protocol. There is a Mentor whose action creates entanglement between the rest of the parties and also determines one of the several possible courses of the communication scheme. After that the Mentor quits. The effect of three different noises, namely, Bit-flip, Phase-flip and Amplitude-damping noises are analyzed using the Kraus operator on the otherwise perfect protocol. The decreased fidelity in the presence of noise is numerically studied with respect to noise and other parameters. It is found that in all the three cases the fidelity tends to one as the noise parameter tends to zero.Detection of the genuine non-locality of any three-qubit statehttps://zbmath.org/1528.810832024-03-13T18:33:02.981707Z"Garg, Anuma"https://zbmath.org/authors/?q=ai:garg.anuma"Adhikari, Satyabrata"https://zbmath.org/authors/?q=ai:adhikari.satyabrataSummary: It is known that the violation of Svetlichny inequality by any three-qubit state described by the density operator \(\rho_{ABC}\) witness the genuine non-locality of \(\rho_{ABC}\). But it is not an easy task as the problem of showing the genuine non-locality of any three-qubit state reduces to the problem of a complicated optimization problem. Thus, the detection of genuine non-locality of any three-qubit state may be considered a challenging task. Therefore, we have taken a different approach and derived the lower and upper bound of the expectation value of the Svetlichny operator with respect to any three-qubit state to study this problem. The expression of the obtained bounds depends on whether the reduced two-qubit entangled state is detected by the CHSH witness operator or not. It may be expressed in terms of the following quantities such as (i) the eigenvalues of the product of the given three-qubit state and the composite system of single qubit maximally mixed state and reduced two-qubit state and (ii) the non-locality of reduced two-qubit state. We then achieve the inequality whose violation may detect the genuine non-locality of any three-qubit state. A few examples are cited to support our obtained results. Lastly, we discuss its possible implementation in the laboratory.Chaos due to symmetry-breaking in deformed Poisson ensemblehttps://zbmath.org/1528.811602024-03-13T18:33:02.981707Z"Das, Adway Kumar"https://zbmath.org/authors/?q=ai:das.adway-kumar"Ghosh, Anandamohan"https://zbmath.org/authors/?q=ai:ghosh.anandamohanSummary: The competition between strength and correlation of coupling terms in a Hamiltonian defines numerous phenomenological models exhibiting spectral properties interpolating between those of Poisson (integrable) and Wigner-Dyson (chaotic) ensembles. It is important to understand how the off-diagonal terms of a Hamiltonian evolve as one or more symmetries of an integrable system are explicitly broken. We introduce a deformed Poisson ensemble to demonstrate an exact mapping of the coupling terms to the underlying symmetries of a Hamiltonian. From the maximum entropy principle we predict a chaotic limit which is numerically verified from the spectral properties and the survival probability calculations.A study of the quasi-probability distributions of the Tavis-Cummings model under different quantum channelshttps://zbmath.org/1528.811652024-03-13T18:33:02.981707Z"Tiwari, Devvrat"https://zbmath.org/authors/?q=ai:tiwari.devvrat"Banerjee, Subhashish"https://zbmath.org/authors/?q=ai:banerjee.subhashish|banerjee.subhashish.1Summary: We study the dynamics of the spin and cavity field of the Tavis-Cummings model using quasi-probability distribution functions and the second-order coherence function, respectively. The effects of (non)-Markovian noise are considered. The relationship between the evolution of the cavity photon number and spin excitation under different quantum channels is observed. The equal-time second-order coherence function is used to study the sub-Poissonian behavior of light and is compared with the two-time second-order coherence function in order to highlight the (anti)-bunching properties of the cavity radiation.The position-momentum commutator as a generalized function: resolution of the apparent discrepancy between continuous and discrete baseshttps://zbmath.org/1528.811702024-03-13T18:33:02.981707Z"Boykin, Timothy B."https://zbmath.org/authors/?q=ai:boykin.timothy-bSummary: It has been known for many years that the matrix representation of the one-dimensional position-momentum commutator calculated with the position and momentum matrices in a finite basis is not proportional to the diagonal matrix, contrary to what one expects from the continuous-space commutator. This discrepancy has correctly been ascribed to the incompleteness of any finite basis, but without the details of exactly why this happens. Understanding why the discrepancy occurs requires calculating the position, momentum, and commutator matrix elements in the continuous position basis, in which all are generalized functions. The reason for the discrepancy is revealed by replacing the generalized functions with sequences approaching them as their parameter approaches zero. Besides explaining the discrepancy in the discrete and continuous models, this investigation finds an unusual double-peaked sequence for the Dirac delta function.Causal perturbative QFT and white noisehttps://zbmath.org/1528.811822024-03-13T18:33:02.981707Z"Wawrzycki, Jarosław"https://zbmath.org/authors/?q=ai:wawrzycki.jaroslawSummary: In this paper, we present the Bogoliubov's causal perturbative QFT, which includes only one refinement: the creation-annihilation operators at a point, i.e. for a specific momentum, are mathematically interpreted as the Hida operators from the white noise analysis. We leave the rest of the theory completely unchanged. This allows avoiding infrared -- and ultraviolet -- divergences in the transition to the adiabatic limit for interacting fields. We present here the analysis of the causal axioms for the scattering operator with the Hida operators as the creation-annihilation operators.Pure quartic wave modulation in optical fiber with the presence of self-steepening and intrapulse Raman scattering responsehttps://zbmath.org/1528.812072024-03-13T18:33:02.981707Z"Tiofack, Camus Gaston Latchio"https://zbmath.org/authors/?q=ai:tiofack.camus-gaston-latchio"Tabi, Conrad Bertrand"https://zbmath.org/authors/?q=ai:tabi.conrad-bertrand"Tagwo, Hippolyte"https://zbmath.org/authors/?q=ai:tagwo.hippolyte"Kofané, Timoléon Crépin"https://zbmath.org/authors/?q=ai:kofane.timoleon-crepinSummary: Modulational instability (MI) is addressed in an optical fiber under competing effects between pure-quartic dispersion (PQD), self-steepening, and intrapulse Raman response. The self-steepening parameter reduces the maximum MI gain and the frequency bandwidth. Under the combined effects of the self-steepening and intrapulse Raman scattering, more spectral windows appear in the gain spectrum, induced by the increasing Raman effect. Numerical results fully concur with the theoretical predictions. The MI is manifested by the emergence of ultrashort pulses trains and rogue wave breathing trains. Under increasing self-steepening effect, asymmetric sidebands, reversible via the Raman scattering effect, appear in the spectral and temporal evolution of the pulse trains, which probes the tunability of energy transfer between the mode during signal propagation. Our results open up further perspectives for exploring mechanisms to generate ultrashort pulses in PQD optical media under higher-order nonlinearities, with applications to silicon photonic crystal waveguides and silica photonic crystal fibers.Consistency of new CDF-II W boson mass with 123-modelhttps://zbmath.org/1528.812182024-03-13T18:33:02.981707Z"Ouazghour, B. Ait"https://zbmath.org/authors/?q=ai:ouazghour.b-ait"Benbrik, R."https://zbmath.org/authors/?q=ai:benbrik.rachid"Ghourmin, E."https://zbmath.org/authors/?q=ai:ghourmin.e"Ouchemhou, M."https://zbmath.org/authors/?q=ai:ouchemhou.m"Rahili, L."https://zbmath.org/authors/?q=ai:rahili.lSummary: Following the recent update measurement of the W boson mass performed by the CDF-II experiment at Fermilab which indicates \(7\sigma\) deviation from the SM prediction. As a consequence, the open question is whether there are extensions of the SM that can carry such a remarkable deviation or what phenomenological repercussions this has. In this paper, we investigate what the theoretical constraints reveal about the 123-model. Also, we study the consistency of a CDF W boson mass measurement with the 123-model expectations, taking into account theoretical and experimental constraints. Both fit results of \(S\) and \(T\) parameters before and after \(m_W^{\mathrm{CDF}}\) measurement are, moreover, considered in this study. Under these conditions, we found that the 123-model prediction is consistent with the measured \(m_W^{\mathrm{CDF}}\) at a 95\% Confidence Level (CL).Anisotropic motion of an electric dipole in a photon gas near a flat conducting boundaryhttps://zbmath.org/1528.812262024-03-13T18:33:02.981707Z"Camargo, G. H. S."https://zbmath.org/authors/?q=ai:camargo.g-h-s"De Lorenci, V. A."https://zbmath.org/authors/?q=ai:de-lorenci.vitorio-a"Ferreira, A. L."https://zbmath.org/authors/?q=ai:ferreira.antonio-luis|ferreira.alex-luiz"Ribeiro, C. C. H."https://zbmath.org/authors/?q=ai:ribeiro.c-c-hSummary: The quantum Brownian motion of a single neutral particle with nonzero electric dipole moment placed in a photon gas at fixed temperature and close to a conducting wall is here examined. The interaction of the particle with the photon field leads to quantum dispersions of its linear and angular momenta, whose magnitudes depend on the temperature, distance to the wall, and also on the dipole moment characteristics. It is shown that for typical experimental parameters the amount of energy held by the dipole rotation is expressively larger than the one related to the center of mass translation. Furthermore, the particle kinetic energy in presence of a thermal bath can decrease if the wall is added to the system, representing a novel quantum cooling effect where the work done by the quantum vacuum extracts energy from the particle. Finally, possible observable consequences are discussed.Fractional quantum Hall effect and M-theoryhttps://zbmath.org/1528.812302024-03-13T18:33:02.981707Z"Vafa, Cumrun"https://zbmath.org/authors/?q=ai:vafa.cumrunSummary: We propose a unifying model for FQHE which on the one hand connects it to recent developments in string theory and on the other hand leads to new predictions for the principal series of experimentally observed FQH systems with filling fraction \(\nu = \frac{n}{2n \pm 1}\) as well as those with \(\nu = \frac{m}{m+2}\). Our model relates these series to minimal unitary models of the Virasoro and super-Virasoro algebra and is based on \(SL(2, \mathbf{C})\) Chern-Simons theory in Euclidean space or \(SL(2, \mathbb{R})\times SL(2, \mathbb{R})\) Chern-Simons theory in Minkowski space. This theory, which has also been proposed as a soluble model for \(2+1\) dimensional quantum gravity, and its \(\mathrm{N} = 1\) supersymmetric cousin, provide effective descriptions of FQHE. The principal series corresponds to quantized levels for the two \(SL(2,\mathbb{R})\)'s such that the diagonal \(SL(2,\mathbb{R})\) has level 1. The model predicts, contrary to standard lore, that for principal series of FQH systems the quasiholes possess non-abelian statistics. For the multi-layer case we propose that complex ADE Chern-Simons theories provide effective descriptions, where the rank of the ADE is mapped to the number of layers. Six dimensional \((2, 0)\) ADE theories on the Riemann surface \(\Sigma\) provides a realization of FQH systems in M-theory. Moreover we propose that the q-deformed version of Chern-Simons theories are related to the anisotropic limit of FQH systems which splits the zeroes of the Laughlin wave function. Extensions of the model to \(3+1\) dimensions, which realize topological insulators with non-abelian topologically twisted Yang-Mills theory is pointed out.Thermal approximation of the equilibrium measure and obstacle problemhttps://zbmath.org/1528.820052024-03-13T18:33:02.981707Z"Armstrong, Scott"https://zbmath.org/authors/?q=ai:armstrong.scott-n"Serfaty, Sylvia"https://zbmath.org/authors/?q=ai:serfaty.sylviaSummary: We consider the probability measure minimizing a free energy functional equal to the sum of a Coulomb interaction, a confinement potential and an entropy term, which arises in the statistical mechanics of Coulomb gases. In the limit where the inverse temperature \(\beta\) tends to \(\infty\) the entropy term disappears and the measure, which we call the ``thermal equilibrium measure'' tends to the well-known equilibrium measure, which can also be interpreted as a solution to the classical obstacle problem. We provide quantitative estimates on the convergence of the thermal equilibrium measure to the equilibrium measure in strong norms in the bulk of the latter, with a sequence of explicit correction terms in powers of \(\beta^{-1}\), as well as an analysis of the tail after the boundary layer of size \(\beta^{-1/2}\).Ergodic property of Langevin systems with superstatistical, uncorrelated or correlated diffusivityhttps://zbmath.org/1528.820352024-03-13T18:33:02.981707Z"Wang, Xudong"https://zbmath.org/authors/?q=ai:wang.xudong"Chen, Yao"https://zbmath.org/authors/?q=ai:chen.yaoSummary: Brownian yet non-Gaussian diffusion has recently been observed in numerous biological and active matter system. The cause of the non-Gaussian distribution have been elaborately studied in the idea of a superstatistical dynamics or a diffusing diffusivity. Based on a random diffusivity model, we here focus on the ergodic property and the scatter of the amplitude of time-averaged mean-squared displacement (TAMSD). By investigating the random diffusivity model with three categories of diffusivities, including diffusivity being a random variable \(D\), a time-dependent but uncorrelated diffusivity \(D(t)\), and a correlated stochastic process \(D(t)\), we find that ensemble-averaged TAMSDs are always normal while ensemble-averaged mean-squared displacement can be anomalous. Further, the scatter of dimensionless amplitude is completely determined by the time average of diffusivity \(D(t)\). Our results are valid for arbitrary diffusivity \(D(t)\).Variable range random walkhttps://zbmath.org/1528.820382024-03-13T18:33:02.981707Z"Odagaki, Takashi"https://zbmath.org/authors/?q=ai:odagaki.takashiSummary: Exploiting the coherent medium approximation, I investigate a random walk on objects distributed randomly in a continuous space when the jump rate depends on the distance between two adjacent objects. In one dimension, it is shown that when the jump rate decays exponentially in the long distance limit, a non-diffusive to diffusive transition occurs as the density of sites is increased. In three dimensions, the transition exists when the jump rate has a super Gaussian decay.Optimization of the Sherrington-Kirkpatrick Hamiltonianhttps://zbmath.org/1528.820432024-03-13T18:33:02.981707Z"Montanari, Andrea"https://zbmath.org/authors/?q=ai:montanari.andrea|montanari.andrea.1Summary: Let \({A}\in{\mathbb R}^{n\times n}\) be a symmetric random matrix with independent and identically distributed (i.i.d.) Gaussian entries above the diagonal. We consider the problem of maximizing \(\langle{\sigma},{A}{\sigma}\rangle\) over binary vectors \({\sigma}\in\{+1,-1\}^n\). In the language of statistical physics, this amounts to finding the ground state of the Sherrington-Kirkpatrick model of spin glasses. The asymptotic value of this optimization problem was characterized by Parisi via a celebrated variational principle, subsequently proved by Talagrand. We give an algorithm that, for any \(\varepsilon>0\), outputs \({\sigma}_*\in\{-1,+1\}^n\) such that \(\langle{\sigma}_*,\boldsymbol{A}{\sigma}_*\rangle\) is at least \((1-\varepsilon)\) of the optimum value, with probability converging to one as \(n\to\infty \). The algorithm's time complexity is \(C(\varepsilon)\, n^2\). We generalize it to matrices with i.i.d., but not necessarily Gaussian, entries, and obtain an algorithm that computes the MAXCUT of a dense Erdös-Renyi random graph to within a factor \((1-\varepsilon\cdot n^{-1/2})\). As a side result, we prove that, at (low) nonzero temperature, the algorithm constructs approximate solutions of the Thouless-Anderson-Palmer equations.Formalism for stochastic perturbations and analysis in relativistic starshttps://zbmath.org/1528.830032024-03-13T18:33:02.981707Z"Satin, Seema"https://zbmath.org/authors/?q=ai:satin.seema-eSummary: Perturbed Einstein's equations with a linear response relation and a stochastic source, applicable to a relativistic star model are worked out. These perturbations which are stochastic in nature, are of significance for building a non-equilibrium statistical mechanics theory in connections with relativistic astrophysics. A fluctuation dissipation relation for a spherically symmetric star in its simplest form is obtained. The FD relation shows how the random velocity fluctuations in the background of the unperturbed star can dissipate into Lagrangian displacement of fluid trajectories of the dense matter. Interestingly in a simple way, a constant (in time) coefficient of dissipation is obtained without a delta correlated noise. This formalism is also extended for perturbed TOV equations which have a stochastic contribution, and show up in terms of the effective or root mean square pressure perturbations. Such contributions can shed light on new ways of analysing the equation of state for dense matter. One may obtain contributions of first and second order in the equation of state using this stochastic approach.Post-Keplerian waveform model for binary compact object as sources of space-based gravitational wave detector and its implicationshttps://zbmath.org/1528.830142024-03-13T18:33:02.981707Z"Li, Li-Fang"https://zbmath.org/authors/?q=ai:li.lifang"Cao, Zhoujian"https://zbmath.org/authors/?q=ai:cao.zhoujianSummary: Binary compact objects will be among the important sources for the future space-based gravitational wave detectors. Such binary compact objects include stellar massive binary black hole, binary neutron star, binary white dwarf and mixture of these compact objects. Regarding to the relatively low frequency, the gravitational interaction between the two objects of the binary is weak. Post-Newtonian approximation of general relativity is valid. Previous works about the waveform model for such binaries in the literature consider the dynamics for specific situations which involve detailed complicated matter dynamics between the two objects. We here take a different idea. We adopt the trick used in pulsar timing detection. For any gravity theories and any detailed complicated matter dynamics, the motion of the binary can always be described as a post-Keplerian expansion. And a post-Keplerian gravitational waveform model will be reduced. Instead of object masses, spins, matter's equation of state parameters and dynamical parameters beyond general relativity, the involved parameters in our post-Keplerian waveform model are the Keplerian orbit elements and their adiabatic variations. Respect to current planning space-based gravitational wave detectors including LISA, Taiji and Tianqin, we find that the involved waveform model parameters can be well determined. And consequently the detail matter dynamics of the binary can be studied then. For binary with purely gravitational interactions, gravity theory can be constrained well.Primordial gravity waves in a rainbow backgroundhttps://zbmath.org/1528.830162024-03-13T18:33:02.981707Z"Salti, M."https://zbmath.org/authors/?q=ai:salti.mustafa|salti.mehmet"Aydogdu, O."https://zbmath.org/authors/?q=ai:aydogdu.oktay|aydogdu.omerSummary: The dawn of the epoch of gravitational wave (GW) astronomy, which initiated with the detection of a fundamental noise, was the period when the search for a theory in which gravity could be quantized began to increase significantly. In this paper, we have mainly intended to focus on the polar modes of GWs in the formalism of gravity's rainbow, which is based on the product of the disparity between quantum mechanics and general relativity (GR). For this purpose, we have perturbed the spatially flat conformal Friedmann-Lemaitre-Robertson-Walker metric, material distribution and the components of four-velocity by making use of the polar Regge-Wheeler gauge and formulated the corresponding field equations for both the zeroth-order (unperturbed) and the first-order (perturbed) cases of the metric. Subsequently, these field equations have been taken into account simultaneously to get exact expressions of the gauge functions. From a graphical perspective, we have studied the impact of rainbow parameters on the amplitude of GWs. In the last step, discussing the Huygens Principle, we have concluded that the GWs obey the principle only in the radiation-dominated era and the principle is broken otherwise.Testing the post-Newtonian expansion with GW170817https://zbmath.org/1528.830172024-03-13T18:33:02.981707Z"Shoom, Andrey A."https://zbmath.org/authors/?q=ai:shoom.andrey-a"Gupta, Pawan K."https://zbmath.org/authors/?q=ai:gupta.pawan-k"Krishnan, Badri"https://zbmath.org/authors/?q=ai:krishnan.badri"Nielsen, Alex B."https://zbmath.org/authors/?q=ai:nielsen.alex-b"Capano, Collin D."https://zbmath.org/authors/?q=ai:capano.collin-dSummary: Observations of gravitational waves from compact binary mergers have enabled unique tests of general relativity in the dynamical and non-linear regimes. One of the most important such tests is constraints on the post-Newtonian (PN) corrections to the phase of the gravitational wave signal. The values of the PN coefficients can be calculated within standard general relativity, and these values are different in many alternate theories of gravity. It is clearly of great interest to constrain the deviations based on gravitational wave observations. In the majority of such tests which have been carried out, and which yield by far the most stringent constraints, it is common to vary these PN coefficients individually. While this might in principle be useful for detecting certain deviations from standard general relativity, it is a serious limitation. For example, we would expect alternate theories of gravity to generically have additional parameters. The corrections to the PN coefficients would be expected to depend on these additional non-GR parameters, whence, we expect that the various PN coefficients to be highly correlated. We present an alternate analysis here using data from the binary neutron star coalescence GW170817. Our analysis uses an appropriate linear combination of non-GR parameters that represent absolute deviations from the corresponding post-Newtonian inspiral coefficients in the TaylorF2 approximant phase. These combinations represent uncorrelated non-GR parameters which correspond to principal directions of their covariance matrix in the parameter subspace. Our results illustrate good agreement with GR. In particular, the integral non-GR phase is \(\Psi_{\tiny int-non-GR} = 0.0447\pm 25.3000\) and the deviation from GR percentile is \(p^{\tiny Dev-GR}_n=25.85\%\).Building spacetime from effective interactions between quantum fluctuationshttps://zbmath.org/1528.830382024-03-13T18:33:02.981707Z"Karlsson, Anna"https://zbmath.org/authors/?q=ai:karlsson.annaSummary: We describe how a model of effective interactions between quantum fluctuations under certain assumptions can be constructed in a way so that the large-scale limit gives an effective theory that matches general relativity (GR) in vacuum regions. This is an investigation of a possible scenario of spacetime emergence from quantum interactions directly in the spacetime, and of how effective quantum behaviour might provide a useful link between detailed properties of quantum interactions and GR. The quantum fluctuations are assumed to entangle sufficiently for a cohesive spacetime to form, so that their effective properties can be described relative to a \(D\)-dimensional reference frame. To obtain the desired features of a smooth metric with a vanishing Ricci tensor, the quantum fluctuations are modelled as Gaussian probability distributions, with a shape set relative to the interactions coming from the surroundings. At small scales, the propagation through the spacetime is modelled by a Gaussian random walk.Constraining neutrino properties and smoothing the Hubble tension via the LSBR modelhttps://zbmath.org/1528.830462024-03-13T18:33:02.981707Z"Dahmani, Safae"https://zbmath.org/authors/?q=ai:dahmani.safae"Bouali, Amine"https://zbmath.org/authors/?q=ai:bouali.amine"El Bojaddaini, Imad"https://zbmath.org/authors/?q=ai:el-bojaddaini.imad"Errahmani, Ahmed"https://zbmath.org/authors/?q=ai:errahmani.ahmed"Ouali, Taoufik"https://zbmath.org/authors/?q=ai:ouali.taoufikSummary: In this paper, we study phantom dark energy (DE) effects on cosmological parameters by considering the different properties of neutrinos. The Little Sibling of the Big Rip (LSBR), which is the phantom DE model under consideration in the current paper induces an abrupt event in the future and deviates from the \(\Lambda\) CDM model by an additional constant parameter. We study different neutrino properties, namely the standard active neutrinos with \(N_{\mathrm{eff}}=3.044\), the standard massive neutrinos, and possible sterile neutrinos with varying \(N_{\mathrm{eff}}\). In the case of standard neutrinos, a slight increase in the \(H_0\) parameter is observed in the LSBR model compared to the \(\Lambda\) CDM model. In the case of massive neutrinos, we notice that LSBR cannot reduce the upper limits on the sum of the neutrino masses but it can increase the value of \(H_0\). Furthermore, in the case of relativistic neutrinos, we obtain \(H_0 = 70.4\pm 0.78\)km s\(^{-1} Mpc^{-1}\) for LSBR model, which reduces the \(H_0\) tension to \(2.03\sigma\).Comparing phantom dark energy models with various diagnostic toolshttps://zbmath.org/1528.830532024-03-13T18:33:02.981707Z"Mhamdi, Dalale"https://zbmath.org/authors/?q=ai:mhamdi.dalale"Bargach, Farida"https://zbmath.org/authors/?q=ai:bargach.farida"Dahmani, Safae"https://zbmath.org/authors/?q=ai:dahmani.safae"Bouali, Amine"https://zbmath.org/authors/?q=ai:bouali.amine"Ouali, Taoufik"https://zbmath.org/authors/?q=ai:ouali.taoufikSummary: In this paper, we apply various diagnostic tools to the phantom dark energy (DE) models, including the big rip (BR), the little sibling of the big rip (LSBR), and the little rip (LR). The evolutionary trajectories of the statefinder \(S_n(z)\), the growth rate \(\epsilon (z)\), and both \(Om (z)\) and \(\omega_d-\omega'_d\) diagnostics are plotted according to the best fit extracted from each model using a Markov Chain Monte Carlo method. In the high-redshift region, all these diagnostics show the degeneracy of the phantom DE models with each other and also with the \(\Lambda\) CDM model, even considering cases of different parameter values shows strong degeneracy. However, in the low-redshift region this degeneracy was broken by using \(S_3^{(1)}\) and \(S_4^{(1)}\) diagnostics, or otherwise by using the composite null diagnostic (\textit{CND}). All diagnostic tools show the rich behavior of the LSBR model as it can be quintessence-like or phantom-like depending on the \(\Omega_{lsbr}\) parameter, in contrast to the LR model which is only phantom-like. A direct comparison of the phantom DE models in the \(\left\{ {S_3^{(1)}, S_4^{(1)}} \right\} , \left\{ {S_3^{(1)}, S_3^{(2)}} \right\} \), and \(\omega_d-\omega'_d\) plane is also made. We further show that the separation between the models can be directly seen at the current state \(\left\{ S_3^{(1)}(t_0),S_4^{(1)}(t_0) \right\}\) of the models. We have reached that these diagnostic tools are rather robust in differentiating not only different DE models but also the same kind of models with different equation of state (EoS) which lead to different fates of the Universe.Data-driven and almost model-independent reconstruction of modified gravityhttps://zbmath.org/1528.830542024-03-13T18:33:02.981707Z"Mu, Yuhao"https://zbmath.org/authors/?q=ai:mu.yuhao"Li, En-Kun"https://zbmath.org/authors/?q=ai:li.en-kun"Xu, Lixin"https://zbmath.org/authors/?q=ai:xu.lixin.1(no abstract)New agegraphic dark energy in Brans-Dicke theory with sign changeable interaction for flat universehttps://zbmath.org/1528.830562024-03-13T18:33:02.981707Z"Pinki"https://zbmath.org/authors/?q=ai:pinki.pinki"Kumar, Pankaj"https://zbmath.org/authors/?q=ai:kumar.pankajSummary: In the present study, we discuss a cosmological model considering interaction between new agegraphic dark energy and dark matter with sign changeable interaction term within the framework of Brans-Dicke theory of gravity for a flat universe. We assume the well motivated logarithmic form of Brans-Dicke scalar field in terms of the scale factor to find the cosmological parameters such as equation of state parameter, deceleration parameter and plot graphs to discuss their evolution against redshift parameter \(z\). It is shown that the equation of state parameter may behave like cosmological constant for suitable values of parameters but it shows quintessence like behavior for different values of model parameters in future. The deceleration parameter shows observationally verified recent phase transition and accelerated expansion of the universe in future. The physical significance of well-known cosmological planes i.e. \(w-w'\) and statefinder diagnostic is also explored for our model. The statefinder diagnostic shows that new agegraphic dark energy behaves like chaplygin gas in early time and behaves like quintessence in future. Moreover, for suitable values of parameters it behaves like cosmological constant at present. The analysis of \(w-w'\) plane shows that our model shows freezing region and reaches in the vicinity of \(\Lambda\)CDM model in future. Further, we apply thermodynamic analysis and found that the generalized second law of thermodynamics is satisfied with in the model.Identifying minimal composite dark matterhttps://zbmath.org/1528.830622024-03-13T18:33:02.981707Z"Xu, Shuai"https://zbmath.org/authors/?q=ai:xu.shuai"Zheng, Sibo"https://zbmath.org/authors/?q=ai:zheng.siboSummary: We attempt to identify the minimal composite scalar dark matter from strong dynamics with the characteristic mass of order TeV scale. We provide direct and indirect limits from dark matter direct detections and collider facilities. Compared to a fundamental scalar dark matter, our results show that in the composite case with sizable derivative interaction between the dark matter and Higgs, the disappearing resonant mass region, the smaller spin-independent dark matter-nucleon scattering cross section in certain dark mass region, and the absence at the HL-LHC provide us an opportunity to distinguish the composite dark matter.Primordial black holes and gravitational waves induced by exponential-tailed perturbationshttps://zbmath.org/1528.830632024-03-13T18:33:02.981707Z"Abe, Katsuya T."https://zbmath.org/authors/?q=ai:abe.katsuya-t"Inui, Ryoto"https://zbmath.org/authors/?q=ai:inui.ryoto"Tada, Yuichiro"https://zbmath.org/authors/?q=ai:tada.yuichiro"Yokoyama, Shuichiro"https://zbmath.org/authors/?q=ai:yokoyama.shuichiro(no abstract)Impact of multiple modes on the evolution of self-interacting axion condensate around rotating black holeshttps://zbmath.org/1528.830812024-03-13T18:33:02.981707Z"Omiya, Hidetoshi"https://zbmath.org/authors/?q=ai:omiya.hidetoshi"Takahashi, Takuya"https://zbmath.org/authors/?q=ai:takahashi.takuya"Tanaka, Takahiro"https://zbmath.org/authors/?q=ai:tanaka.takahiro"Yoshino, Hirotaka"https://zbmath.org/authors/?q=ai:yoshino.hirotaka(no abstract)Gravitational waves from eccentric extreme mass-ratio inspirals as probes of scalar fieldshttps://zbmath.org/1528.830922024-03-13T18:33:02.981707Z"Zhang, Chao"https://zbmath.org/authors/?q=ai:zhang.chao.5|zhang.chao.1|zhang.chao|zhang.chao.17|zhang.chao.11|zhang.chao.3|zhang.chao.2|zhang.chao.12"Gong, Yungui"https://zbmath.org/authors/?q=ai:gong.yungui"Liang, Dicong"https://zbmath.org/authors/?q=ai:liang.dicong"Wang, Bin"https://zbmath.org/authors/?q=ai:wang.bin.1(no abstract)Detecting vector charge with extreme mass ratio inspirals onto Kerr black holeshttps://zbmath.org/1528.830932024-03-13T18:33:02.981707Z"Zhang, Chao"https://zbmath.org/authors/?q=ai:zhang.chao.1|zhang.chao.3|zhang.chao.12|zhang.chao.17|zhang.chao.11|zhang.chao|zhang.chao.5|zhang.chao.2"Guo, Hong"https://zbmath.org/authors/?q=ai:guo.hong"Gong, Yungui"https://zbmath.org/authors/?q=ai:gong.yungui"Wang, Bin"https://zbmath.org/authors/?q=ai:wang.bin.1(no abstract)Ricci fall-off in static and stationary non-singular spacetimes, revisited: the null geodesic methodhttps://zbmath.org/1528.830972024-03-13T18:33:02.981707Z"Garfinkle, David"https://zbmath.org/authors/?q=ai:garfinkle.david"Harris, Stacey G."https://zbmath.org/authors/?q=ai:harris.stacey-gSummary: We revisit an older concept in singularity theory: that in the presence of the strong energy condition (SEC), a static or stationary spacetime must have a quadratic fall-off in a characteristic Ricci quantity, in order for the spacetime to be without singularities (or, at least, to be both globally hyperbolic and timelike or null geodesically complete). We replace SEC with the null energy condition (NEC), and apply the methods used previously on timelike geodesics, to null geodesics instead. The results are noticeably weaker for the NEC case than for SEC: using a somewhat different characteristic measure of Ricci curvature, we obtain a fall-off which is quadratic only if there is not much asymptotic change in the size of the Killing field: we employ a ratio of maximum size to square of minimum size of the Killing field -- within a ball of given radius \(r\) -- in addition to \(1/r^2\).Non-minimal coupling inflation and dark matter under the \(\mathbb{Z}_3\) symmetryhttps://zbmath.org/1528.831072024-03-13T18:33:02.981707Z"Cheng, Wei"https://zbmath.org/authors/?q=ai:cheng.wei"Liu, Xuewen"https://zbmath.org/authors/?q=ai:liu.xuewen"Zhou, Ruiyu"https://zbmath.org/authors/?q=ai:zhou.ruiyu(no abstract)Model-independent bubble wall velocities in local thermal equilibriumhttps://zbmath.org/1528.831192024-03-13T18:33:02.981707Z"Ai, Wen-Yuan"https://zbmath.org/authors/?q=ai:ai.wen-yuan"Laurent, Benoit"https://zbmath.org/authors/?q=ai:laurent.benoit"van de Vis, Jorinde"https://zbmath.org/authors/?q=ai:van-de-vis.jorinde(no abstract)An analytical study of the primordial gravitational-wave-induced contribution to the large-scale structure of the universehttps://zbmath.org/1528.831242024-03-13T18:33:02.981707Z"Bari, Pritha"https://zbmath.org/authors/?q=ai:bari.pritha"Bertacca, Daniele"https://zbmath.org/authors/?q=ai:bertacca.daniele"Bartolo, Nicola"https://zbmath.org/authors/?q=ai:bartolo.nicola"Ricciardone, Angelo"https://zbmath.org/authors/?q=ai:ricciardone.angelo"Giardiello, Serena"https://zbmath.org/authors/?q=ai:giardiello.serena"Matarrese, Sabino"https://zbmath.org/authors/?q=ai:matarrese.sabino(no abstract)Minimal decoherence from inflationhttps://zbmath.org/1528.831302024-03-13T18:33:02.981707Z"Burgess, C. P."https://zbmath.org/authors/?q=ai:burgess.clifford-p|burgess.cliff-p"Holman, R."https://zbmath.org/authors/?q=ai:holman.r-a|holman.richard"Kaplanek, Greg"https://zbmath.org/authors/?q=ai:kaplanek.greg"Martin, Jérôme"https://zbmath.org/authors/?q=ai:martin.jerome"Vennin, Vincent"https://zbmath.org/authors/?q=ai:vennin.vincent(no abstract)Nonminimal derivative coupling cosmology and the speed of gravitational waveshttps://zbmath.org/1528.831672024-03-13T18:33:02.981707Z"Torres, Isaac"https://zbmath.org/authors/?q=ai:torres.isaac"de Melo Santos, Felipe"https://zbmath.org/authors/?q=ai:de-melo-santos.felipeSummary: The so-called Nonminimal Derivative Coupling (NDC) is an alternative to General Relativity, which produces an asymptotic inflationary mechanism when applied to cosmology. The detection of gravitational waves in the last decade has imposed very stringent constraints over gravitational theories, which gave rise to a massive revision of those theories, in order to investigate the compatibility between them and that observational data. In this paper, we review NDC and address the question if it is compatible with gravitational waves or not. We show that the very existence of gravitational waves in this theory is restricted to a limited range in phase space and there are no accelerated solutions compatible with the present day data for the speed of such waves. This last result is alleviated by the fact that we did not detect primordial gravitational waves so far. Those conclusions are based on the comparison between the expression for the speed of tensor perturbations and the phase space. Finally, some possible scenarios and solutions are considered.Model-dependent analysis method for energy budget of the cosmological first-order phase transitionhttps://zbmath.org/1528.831722024-03-13T18:33:02.981707Z"Wang, Xiao"https://zbmath.org/authors/?q=ai:wang.xiao.6|wang.xiao.5|wang.xiao"Tian, Chi"https://zbmath.org/authors/?q=ai:tian.chi"Huang, Fa Peng"https://zbmath.org/authors/?q=ai:huang.fa-peng(no abstract)Quantum mechanical calculations of synchro-curvature radiations: maser possibilityhttps://zbmath.org/1528.850092024-03-13T18:33:02.981707Z"Tomoda, Hiroko"https://zbmath.org/authors/?q=ai:tomoda.hiroko"Yamada, Shoichi"https://zbmath.org/authors/?q=ai:yamada.shoichiSummary: We calculate the radiative transition rates for synchro-curvature radiation to explore the possibility of maser in the environment that may occur in the magnetosphere of neutron stars (NSs). Unlike previous studies, we employ relativistic quantum mechanics, solving the Dirac equation for an electron in helical magnetic fields. Following \textit{G. Voisin} et al. [Phys. Rev. D (3) 95, No. 8, Article ID 085002, 16 p. (2017, \url{doi:10.1103/PhysRevD.95.085002}); Phys. Rev. D (3) 95, No. 10, Article ID 105008, 17 p. (2017; \url{doi:10.1103/PhysRevD.95.105008})], we utilize adiabatic spinor rotations, under the assumption that the curvature of magnetic-field lines is much larger than the Larmor radius, to obtain the wave functions of an electron. We classify the electron states either by the spin operator projected on the magnetic field or by the helicity operator. To demonstrate that there is a regime where the true absorption rate becomes negative, we numerically evaluate the obtained formulae for some parameter values that may be encountered in the outer gaps of different types of NSs. We show that there is indeed a range of parameters for the negative true absorption rate to occur. We will also study the dependence on those parameters systematically and discuss the classical limit of our formulae. We finally give a crude estimate of the amplification factor in the same environment.Characterization theorems for pseudo cross-variogramshttps://zbmath.org/1528.860082024-03-13T18:33:02.981707Z"Dörr, Christopher"https://zbmath.org/authors/?q=ai:dorr.christopher"Schlather, Martin"https://zbmath.org/authors/?q=ai:schlather.martinSummary: Pseudo cross-variograms appear naturally in the context of multivariate Brown-Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo cross-variogram, and further provide a Schoenberg-type result connecting pseudo cross-variograms and multivariate correlation functions. By means of these characterizations, we provide extensions of the popular univariate space-time covariance model of Gneiting to the multivariate case.Iterated tour partitioning for Euclidean capacitated vehicle routinghttps://zbmath.org/1528.900402024-03-13T18:33:02.981707Z"Mathieu, Claire"https://zbmath.org/authors/?q=ai:mathieu.claire"Zhou, Hang"https://zbmath.org/authors/?q=ai:zhou.hangSummary: We give a probabilistic analysis of the unit-demand Euclidean capacitated vehicle routing problem in the random setting. The objective is to visit all customers using a set of routes of minimum total length, such that each route visits at most \(k\) customers. The best known polynomial-time approximation is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the solution obtained by the ITP algorithm is arbitrarily close to the optimum when \(k\) is either \(o(\sqrt{n})\) or \(\omega (\sqrt{n})\), and they asked whether the ITP algorithm was ``also effective in the intermediate range''. In this work, we show that the ITP algorithm is at best a \((1+{c}_0)\)-approximation, for some positive constant \({c}_0\), and is at worst a 1.915-approximation.
{{\copyright} 2022 Wiley Periodicals LLC}Autonomous traffic at intersections: an optimization-based analysis of possible time, energy, and CO\({}_2\) savingshttps://zbmath.org/1528.900702024-03-13T18:33:02.981707Z"Le, Do Duc"https://zbmath.org/authors/?q=ai:le.do-duc"Merkert, Maximilian"https://zbmath.org/authors/?q=ai:merkert.maximilian"Sorgatz, Stephan"https://zbmath.org/authors/?q=ai:sorgatz.stephan"Hahn, Mirko"https://zbmath.org/authors/?q=ai:hahn.mirko"Sager, Sebastian"https://zbmath.org/authors/?q=ai:sager.sebastianSummary: In the field of autonomous driving, traffic-light-controlled intersections are of special interest. We analyze how much an optimized coordination of vehicles and infrastructure can contribute to efficient transit through these bottlenecks, depending on traffic density and certain regulations of traffic lights. To this end, we develop a mixed-integer linear programming model to describe the interaction between traffic lights and discretized traffic flow. It is based on a microscopic traffic model with centrally controlled autonomous vehicles. We aim to determine a globally optimal traffic flow for given scenarios on a simple, but extensible, urban road network. The resulting models are very challenging to solve, in particular when involving additional realistic traffic-light regulations such as minimum red and green times. While solving times exceed real-time requirements, our model allows an estimation of the maximum performance gains due to improved communication and serves as a benchmark for heuristic and decentralized approaches.
{{\copyright} 2021 The Authors. \textit{Networks} published by Wiley Periodicals LLC.}Equilibrium behavior in tandem Markovian queues with heterogeneous delay-sensitive customershttps://zbmath.org/1528.900782024-03-13T18:33:02.981707Z"Dimitrakopoulos, Yiannis"https://zbmath.org/authors/?q=ai:dimitrakopoulos.yiannisSummary: We consider a system of two unobservable Markovian queues in tandem with strategic customers, who are heterogeneous regarding their delay sensitivity. The customers decide upon arrival whether to balk or join the system and receive service in the first queue or in both queues. We analyze their equilibrium strategic behavior which is specified by double-threshold strategies regarding their delay sensitivity parameter (one threshold for each queue). Moreover, we compare the strategic behavior of the heterogeneous customer population with its homogeneous counterpart. We complement our theoretical results with numerical experiments and provide managerial insights into the optimal control of the system parameters.Average cost minimization in a multi-server retrial queueing system with a controllable reserve group of servershttps://zbmath.org/1528.900792024-03-13T18:33:02.981707Z"Efrosinin, Dmitry"https://zbmath.org/authors/?q=ai:efrosinin.dmitry"Stepanova, Natalia"https://zbmath.org/authors/?q=ai:stepanova.natalia-aSummary: The paper deals with a dynamic optimal activation and deactivation of a reserve group of servers in a multi-server retrial queueing system. The repeated attempts to occupy the server from the orbit occur due to a classical retrial discipline, and the activation of reserves requires a random set-up time. The optimal policy minimizes the average cost per unit of time which consists of the holding and usage cost components. The optimal control policy belongs to a class of hysteretic control policies defined by two threshold levels for the orbit size and prescribes the necessity to activate and deactivate a standby reserve group of servers. A dynamic-programming approach for the Markov decision process is used to show that the optimal thresholds depend on the number of busy servers as well. In order to provide a less computationally expensive method for calculating optimal thresholds, explicit heuristic expressions are proposed to guarantee at least a quasi-optimal solution to the optimization problem. Numerical examples confirm the efficiency of the proposed approach.
For the entire collection see [Zbl 1517.68029].On the distribution of the number of consecutively lost customers in the \textit{BMAP}/\textit{PH}/1/\textit{N} systemhttps://zbmath.org/1528.900832024-03-13T18:33:02.981707Z"Klimenok, Valentina"https://zbmath.org/authors/?q=ai:klimenok.valentina-ivanovna"Dudin, Alexander"https://zbmath.org/authors/?q=ai:dudin.alexander-nSummary: In this paper, we propose method for calculating the distribution of the number of consecutively lost customers in the single-server queueing system with a finite buffer, batch Markovian arrival process and phase type distribution of service time. The most well-known and important performance measure of finite capacity systems is the probability of losing an arbitrary customer. Loss probability is the subject of research in the literature under various assumptions about the nature of the input flow and the distribution of service time. At the same time, this characteristic may be not always a good estimate of the quality of service in queuing systems that arise in the mathematical modeling of telecommunication networks. More indicative in this case is the probability of losing several customers in a row caused by an overflowing buffer. We propose explicit formulas that characterize the distribution and mathematical expectation of the number of consecutively lost customers in the system under consideration.
For the entire collection see [Zbl 1517.68029].Verification of stability condition in unreliable two-class retrial system with constant retrial rateshttps://zbmath.org/1528.900852024-03-13T18:33:02.981707Z"Nekrasova, Ruslana"https://zbmath.org/authors/?q=ai:nekrasova.ruslana"Morozov, Evsey"https://zbmath.org/authors/?q=ai:morozov.evsey"Efrosinin, Dmitry"https://zbmath.org/authors/?q=ai:efrosinin.dmitrySummary: A two-class single-server retrial system with Poisson inputs is considered. In this system, unlike conventional retrial systems, each new ith class customer joins the `end' of a virtual \(i\)th class orbit, and the `oldest' customer from each orbit is only allowed to make an attempt to occupy server after a class-dependent exponential retrial time. Moreover, the server is assumed to be not reliable, and a customer whose service is interrupted joins the `top' of class-\(i\) orbit queue. Thus FIFO discipline is applied in both orbits. Using regenerative methodology and Markov Chain approach we derive stability conditions of this system relying on analysis for less-complicated model with reliable server. Obtained conditions are verified by simulation. Additionally, we analyze a controllable variant of the main model operating under a \(c\mu \)-rule. For that case the system becomes less stable comparing to the non-controllable counterpart.
For the entire collection see [Zbl 1517.68029].On mixed censored \(\delta \)-shock modelshttps://zbmath.org/1528.900892024-03-13T18:33:02.981707Z"Chadjiconstantinidis, Stathis"https://zbmath.org/authors/?q=ai:chadjiconstantinidis.stathisSummary: In this paper we introduce the mixed censored \(\delta \)-shock model which combines the censored \(\delta \)-shock model and the classical extreme shock model. Under the mixed censored \(\delta \)-shock model, the system fails whenever no shock occurs within a \(\delta \)-length time period from the last shock, or the magnitude of the shock is larger than another critical threshold \(\gamma > 0\). For the discrete-time case of occurrence of shocks, by assuming the dependence between intershock times and the corresponding magnitudes of shocks we derive the probability generating function (pgf) of the lifetime of the system, and a matrix-based expression is obtained for the exact distribution of the system's lifetime when the distribution of intershock times and the magnitudes of shocks have a discrete bivariate phase-type distribution. Similar results are obtained by assuming the independence between intershock times and the corresponding magnitudes of shocks, and by proving that the distribution of the shifted lifetime at \(\delta \), is the convolution of a discrete compound geometric distribution and a discrete compound Bernoulli distribution, we get several results concerning the distribution of system's lifetime, like as, simple and efficient recursions for evaluating the survival function and the probability mass function (pmf), the mean and the variance of system's lifetime, as well as discrete Lundberg-type upper bounds for the reliability function. For the continuous-time case of occurrence of shocks we obtain an exact formula for the reliability function and the Laplace-Stieltjes transform of system's lifetime by assuming the dependence between intershock times and the corresponding magnitudes of shocks, and under the independence setup, we obtain the reliability function when the intershock times have the uniform distribution, and we give an asymptotic result under the Poisson process for the arrival of shocks. Similar to the discrete-time case, it is shown that the distribution of the shifted lifetime at \(\delta \), is the convolution of a compound geometric distribution and a compound Bernoulli distribution and using this we obtain two-sided Lundberg-type bounds for the survival function. Finally, some numerical examples to illustrate our results, are also given.Generalized non-renewing replacement warranty policy and an age-based post-warranty maintenance strategyhttps://zbmath.org/1528.900922024-03-13T18:33:02.981707Z"Liu, Peng"https://zbmath.org/authors/?q=ai:liu.peng.3"Wang, Guanjun"https://zbmath.org/authors/?q=ai:wang.guanjunSummary: Nowadays, to protect the interests of customers, quite a few warranted products are sold with some kinds of repair thresholds. For example, a mobile phone manufacturer may provide a replacement service if the phone sold still cannot work after several repair attempts or the cumulative repair time reaches 7 days or 1 month. Motivated by this observation, we study a non-renewing replacement warranty policy with a repair number threshold and a repair time threshold, in which the manufacturer has responsibility for replacing a product when the number of failures is beyond the repair number threshold or the cumulative repair time reaches the repair time threshold, whichever occurs first. In addition, an age-based post-warranty maintenance strategy is developed for the non-renewing replacement warranted product, which provides a lower cost rate than the existing ones. Based on the indicator function method and renewal process theory, we derive the product's life cycle cost rate. Finally, considerable numerical experiments are carried out to show the influences of various parameters on the proposed maintenance strategy.Optimal preventive maintenance policies for products with multiple failure modes after geometric warranty expiryhttps://zbmath.org/1528.900932024-03-13T18:33:02.981707Z"Liu, Peng"https://zbmath.org/authors/?q=ai:liu.peng.3"Wang, Guanjun"https://zbmath.org/authors/?q=ai:wang.guanjunSummary: In this paper, the optimal maintenance strategy for the warranty product subject to multiple failure modes and repair time limit is studied from the customer's perspective. When the product fails, a minimal repair based on the failure type is conducted. If the product can not be repaired within a given time limit, it will be replaced with a new one by the manufacturer during the warranty period. Different from the traditional renewing warranty policy, a geometric warranty policy is adopted in this work, under which once the warranty product is replaced; the length of warranty period is geometrically renewed. After the warranty expires, the product undergoes preventive maintenance (PM) periodically and the PM effect is modeled by reducing the virtual age of different failure modes by different degree. The objective is to determine the optimal interval time for PM and the optimal number of PMs by minimizing the expected life cycle cost rate of product. The existence and uniqueness of the optimal PM policy are proved theoretically, and the optimal bivariate policy can be obtained by using a recursive seeking algorithm. Numerical examples illustrate the effectiveness of geometric warranty policy and the proposed PM model.Optimal task-driven time-dependent covariate-based maintenance policyhttps://zbmath.org/1528.900942024-03-13T18:33:02.981707Z"Misaii, Hasan"https://zbmath.org/authors/?q=ai:misaii.hasan"Fouladirad, Mitra"https://zbmath.org/authors/?q=ai:fouladirad.mitra"Haghighi, Firoozeh"https://zbmath.org/authors/?q=ai:haghighi.firoozehSummary: In this paper, a multi-component series system is considered. The system is monitored periodically. The exact cause of failure is assumed to be masked or missed. In the masked setup, the exact cause of failure is unknown but the set to which it belongs, called the masked set, is known. While in the missing setup, there is no information about the exact cause of failure. A time-dependent covariate-based maintenance policy is proposed such that the maintenance action and cost of the failed components at inspection times depend on several factors and can vary. The component lifetime distributions are considered unknown. The proposed maintenance policy is optimized using some task-driven decision-making statistical learning methods. Finally, the applicability of the proposed theory is analyzed through some numerical analysis. The results are compared to the case where lifetime distributions are known as a benchmark.Evaluation of the accident rate of a plant equipped with an aging single protective channel by the method of supplementary variableshttps://zbmath.org/1528.900952024-03-13T18:33:02.981707Z"Oliveira, L. G."https://zbmath.org/authors/?q=ai:oliveira.l-g-s"Teixeira, D. G."https://zbmath.org/authors/?q=ai:teixeira.d-g"Melo, P. F. Frutuoso e"https://zbmath.org/authors/?q=ai:melo.p-f-frutuoso-e(no abstract)Stochastic comparisons of relevation allocation policies in coherent systemshttps://zbmath.org/1528.900982024-03-13T18:33:02.981707Z"Zhang, Jiandong"https://zbmath.org/authors/?q=ai:zhang.jiandong"Zhang, Yiying"https://zbmath.org/authors/?q=ai:zhang.yiyingSummary: In reliability engineering, the relevation model can be adopted to characterize the performance of redundancy allocation for coherent systems. In this paper, we investigate the allocation problems of relevations for two nodes in a coherent system with independent components for enhancing system reliability. We first investigate the optimal allocation policy of two relevations for two nodes of the system under certain conditions. As a special setting of the relevation, we further discuss optimal allocation strategies for a batch of minimal repairs allocated to two components of the coherent system by applying the useful tool of majorization order. Sufficient conditions are established in terms of structural relationships between the components induced by minimal cut or path sets and the reliabilities of components and relevations. Some numerical examples are provided as illustrations. A real application in aircraft indicator lights systems is also presented to show the availability of our results.Effects of prioritized input on human resource control in departmentalized Markov manpower frameworkhttps://zbmath.org/1528.901302024-03-13T18:33:02.981707Z"Ossai, E. O."https://zbmath.org/authors/?q=ai:ossai.everestus-o"Madukaife, M. S."https://zbmath.org/authors/?q=ai:madukaife.mbanefo-s"Udom, A. U."https://zbmath.org/authors/?q=ai:udom.akaninyene-udo"Nduka, U. C."https://zbmath.org/authors/?q=ai:nduka.uchenna-chinedu"Ugah, T. E."https://zbmath.org/authors/?q=ai:ugah.tobias-ejioforSummary: In this paper, extended Markov manpower models are formulated by incorporating a new class of members of a departmentalized manpower system in a homogeneous Markov manpower model. The new class, called limbo class, admits members of the system who exit to a limbo state for possible re-engagement in the active class. This results to two channels of recruitment: one from the limbo class and another from the outside environment. The idea is motivated by the need to preserve trained and experienced individuals who could be lost in times of financial crises or due to contract completion. The control aspect of the manpower structure under the extended models are examined. Under suitable stochastic condition for the flow matrices, it is proved that the maintainability of the manpower structure through promotion does not depend on the structural form of the limbo class when the system is expanding with priority on recruitment from outside environment, nor on the structural form of the active class when the system is shrinking with priority on recruitment from the limbo class. Necessary and sufficient conditions for maintainability of the manpower structure through recruitment in the case of expanding systems are also established with proofs.Mean field games with absorption and common noise with a model of bank runhttps://zbmath.org/1528.910082024-03-13T18:33:02.981707Z"Burzoni, Matteo"https://zbmath.org/authors/?q=ai:burzoni.matteo"Campi, Luciano"https://zbmath.org/authors/?q=ai:campi.lucianoSummary: We consider a mean field game describing the limit of a stochastic differential game of \(N\)-players whose state dynamics are subject to idiosyncratic and common noise and that can be absorbed when they hit a prescribed region of the state space. We provide a general result for the existence of weak mean field equilibria which, due to the absorption and the common noise, are given by random flow of sub-probabilities. We first use a fixed point argument to find solutions to the mean field problem in a reduced setting resulting from a discretization procedure and then we prove convergence of such equilibria to the desired solution. We exploit these ideas also to construct \(\varepsilon\)-Nash equilibria for the \(N\)-player game. Since the approximation is two-fold, one given by the mean field limit and one given by the discretization, some suitable convergence results are needed. We also introduce and discuss a novel model of bank run that can be studied within this framework.Hedging portfolio for a market model of degenerate diffusionshttps://zbmath.org/1528.910652024-03-13T18:33:02.981707Z"Çağlar, Mine"https://zbmath.org/authors/?q=ai:caglar.mine"Demirel, İhsan"https://zbmath.org/authors/?q=ai:demirel.ihsan"Üstünel, Ali Süleyman"https://zbmath.org/authors/?q=ai:ustunel.ali-suleymanSummary: We consider a semimartingale market model when the underlying diffusion has a singular volatility matrix and compute the hedging portfolio for a given payoff function. Recently, the representation problem for such degenerate diffusions as a stochastic integral with respect to a martingale has been completely settled. This representation and Malliavin calculus established further for the functionals of a degenerate diffusion process constitute the basis of the present work. Using the Clark-Hausmann-Bismut-Ocone type representation formula derived for these functionals, we prove a version of this formula under an equivalent martingale measure. This allows us to derive the hedging portfolio as a solution of a system of linear equations. The uniqueness of the solution is achieved by a projection idea that lies at the core of the martingale representation at the first place. We demonstrate the hedging strategy as explicitly as possible with some examples of the payoff function such as those used in exotic options, whose value at maturity depends on the prices over the entire time horizon.Asymptotic minimization of expected time to reach a large wealth level in an asset market gamehttps://zbmath.org/1528.910722024-03-13T18:33:02.981707Z"Zhitlukhin, Mikhail"https://zbmath.org/authors/?q=ai:zhitlukhin.mikhail-vSummary: A stochastic game-theoretic model of a discrete-time asset market with short-lived assets and endogenous asset prices is considered. It is proved that the strategy which invests in the assets proportionally to their expected relative payoffs asymptotically minimizes the expected time needed to reach a large wealth level under the assumption that the total payoffs of the assets are i.i.d. and the relative payoffs are bounded away from zero.Pricing Bermudan options using regression trees/random forestshttps://zbmath.org/1528.910732024-03-13T18:33:02.981707Z"Ech-Chafiq, Zineb El Filali"https://zbmath.org/authors/?q=ai:el-filali-ech-chafiq.zineb"Labordère, Pierre Henry"https://zbmath.org/authors/?q=ai:henry-labordere.pierre"Lelong, Jérôme"https://zbmath.org/authors/?q=ai:lelong.jeromeSummary: The value of an American option is the maximized value of the discounted cash flows from the option. At each time step, one needs to compare the immediate exercise value with the continuation value and decide to exercise as soon as the exercise value is strictly greater than the continuation value. We can formulate this problem as a dynamic programming equation, where the main difficulty comes from the computation of the conditional expectations representing the continuation values at each time step. In [\textit{F. A. Longstaff} and \textit{E. S. Schwartz}, Rev. Financ. Stud. 6, No. 2, 327--343 (1993; Zbl 1386.91144)], these conditional expectations were estimated using regressions on a finite-dimensional vector space (typically a polynomial basis). In this paper, we follow the same algorithm; only the conditional expectations are estimated using regression trees or random forests. We discuss the convergence of the Longstaff and Schwartz algorithm when the standard least squares regression is replaced by regression trees. Finally, we expose some numerical results with regression trees and random forests. The random forest algorithm gives excellent results in high dimensions.A contagion process with self-exciting jumps in credit risk applicationshttps://zbmath.org/1528.910772024-03-13T18:33:02.981707Z"Pasricha, Puneet"https://zbmath.org/authors/?q=ai:pasricha.puneet"Selvamuthu, Dharmaraja"https://zbmath.org/authors/?q=ai:dharmaraja.selvamuthu|selvamuthu.dharmaraja"Natarajan, Selvaraju"https://zbmath.org/authors/?q=ai:natarajan.selvarajuSummary: The modeling of the probability of joint default or total number of defaults among the firms is one of the crucial problems to mitigate the credit risk since the default correlations significantly affect the portfolio loss distribution and hence play a significant role in allocating capital for solvency purposes. In this article, we derive a closed-form expression for the default probability of a single firm and probability of the total number of defaults by time \(t\) in a homogeneous portfolio. We use a contagion process to model the arrival of credit events causing the default and develop a framework that allows firms to have resistance against default unlike the standard intensity-based models. We assume the point process driving the credit events is composed of a systematic and an idiosyncratic component, whose intensities are independently specified by a mean-reverting affine jump-diffusion process with self-exciting jumps. The proposed framework is competent of capturing the feedback effect. We further demonstrate how the proposed framework can be used to price synthetic collateralized debt obligation (CDO). Finally, we present the sensitivity analysis to demonstrate the effect of different parameters governing the contagion effect on the spread of tranches and the expected loss of the CDO.Error analysis of finite difference scheme for American option pricing under regime-switching with jumpshttps://zbmath.org/1528.910782024-03-13T18:33:02.981707Z"Huang, Cunxin"https://zbmath.org/authors/?q=ai:huang.cunxin"Song, Haiming"https://zbmath.org/authors/?q=ai:song.haiming"Yang, Jinda"https://zbmath.org/authors/?q=ai:yang.jinda"Zhou, Bocheng"https://zbmath.org/authors/?q=ai:zhou.bochengThis paper proposes an efficient numerical method for evaluating American options under regime-switching jump-diffusion models (Merton's and Kou's models). By the relation of optimal exercise boundaries among several options, a simplified model defined on a bounded domain is first presented to approximate the original model defined on an unbounded domain. Then a composite trapezoidal formula is applied, which guarantees that the integral discretized matrix is a Toeplitz matrix. More precisely, a finite difference method are applied to discretize the simplified model to be an LCP in finite dimensional space. Simultaneously, the authors analyze the related properties of the discretization scheme and estimate the convergence error. Finally, several numerical simulations are carried out to verify the proposed method's theoretical analysis and efficiency.
Reviewer: Nikolay Kyurkchiev (Plovdiv)Computable error bounds of Laplace inversion for pricing Asian optionshttps://zbmath.org/1528.910802024-03-13T18:33:02.981707Z"Song, Yingda"https://zbmath.org/authors/?q=ai:song.yingda"Cai, Ning"https://zbmath.org/authors/?q=ai:cai.ning"Kou, Steven"https://zbmath.org/authors/?q=ai:kou.stevenSummary: The prices of Asian options, which are among the most important options in financial engineering, can often be written in terms of Laplace transforms. However, computable error bounds of the Laplace inversions are rarely available to guarantee their accuracy. We conduct a thorough analysis of the inversion of the Laplace transforms for continuously and discretely monitored Asian option prices under general continuous-time Markov chains (CTMCs), which can be used to approximate any one-dimensional Markov process. More precisely, we derive computable bounds for the discretization and truncation errors involved in the inversion of Laplace transforms. Numerical results indicate that the algorithm is fast and easy to implement, and the computable error bounds are especially suitable to provide benchmark prices under CTMCs.A periodic averaging method for impulsive stochastic age-structured population model in a polluted environmenthttps://zbmath.org/1528.920282024-03-13T18:33:02.981707Z"Li, Wenrui"https://zbmath.org/authors/?q=ai:li.wenrui"Ye, Ming"https://zbmath.org/authors/?q=ai:ye.ming"Zhang, Qimin"https://zbmath.org/authors/?q=ai:zhang.qimin"Anke, Meyer-Baese"https://zbmath.org/authors/?q=ai:anke.meyer-baese"Li, Yan"https://zbmath.org/authors/?q=ai:li.yan.58(no abstract)Data-driven models for the risk of infection and hospitalization during a pandemic: case study on COVID-19 in Nepalhttps://zbmath.org/1528.920322024-03-13T18:33:02.981707Z"Adhikari, Khagendra"https://zbmath.org/authors/?q=ai:adhikari.khagendra"Gautam, Ramesh"https://zbmath.org/authors/?q=ai:gautam.ramesh-s"Pokharel, Anjana"https://zbmath.org/authors/?q=ai:pokharel.anjana"Uprety, Kedar Nath"https://zbmath.org/authors/?q=ai:uprety.kedar-nath"Vaidya, Naveen K."https://zbmath.org/authors/?q=ai:vaidya.naveen-kSummary: The newly emerging pandemic disease often poses unexpected troubles and hazards to the global health system, particularly in low and middle-income countries like Nepal. In this study, we developed mathematical models to estimate the risk of infection and the risk of hospitalization during a pandemic which are critical for allocating resources and planning health policies. We used our models in Nepal's unique data set to explore national and provincial-level risks of infection and risk of hospitalization during the Delta and Omicron surges. Furthermore, we used our model to identify the effectiveness of non-pharmaceutical interventions (NPIs) to mitigate COVID-19 in various groups of people in Nepal. Our analysis shows no significant difference in reproduction numbers in provinces between the Delta and Omicron surge periods, but noticeable inter-provincial disparities in the risk of infection (for example, during Delta (Omicron) surges, the risk of infection of Bagmati province is: \(\sim 98.94 (89.62)\); Madhesh province: \(\sim 12.16 (5.1)\); Karnali province \(\sim 31.16 (3)\) per hundred thousands). Our estimates show a significantly low level of hospitalization risk during the Omicron surge compared to the Delta surge (hospitalization risk is: \(\sim 10\%\) in Delta and \(\sim 2.5\%\) in Omicron). We also found significant inter-provincial disparities in the hospitalization rate (for example, \(\sim 6\%\) in Madhesh province and \(\sim 21\%\) in Sudur Paschim) during the Delta surge. Moreover, our results show that closing only schools, colleges, and workplaces reduces the risk of infection by one-third, while a complete lockdown reduces the infections by two-thirds. Our study provides a framework for the computation of the risk of infection and the risk of hospitalization and offers helpful information for controlling the pandemic.A note about the invariance of the basic reproduction number for stochastically perturbed SIS modelshttps://zbmath.org/1528.920352024-03-13T18:33:02.981707Z"Bernardi, Enrico"https://zbmath.org/authors/?q=ai:bernardi.enrico"Lanconelli, Alberto"https://zbmath.org/authors/?q=ai:lanconelli.albertoSummary: In [\textit{A. Gray} et al., SIAM J. Appl. Math. 71, No. 3, 876--902 (2011; Zbl 1263.34068)] a susceptible-infected-susceptible (SIS) stochastic differential equation (SDE), obtained via a suitable random perturbation of the disease transmission coefficient in the classic SIS model, has been studied. Such random perturbation enters via an informal manipulation of stochastic differentials and leads to an Itô's-type SDE. The authors identify a stochastic reproduction number, which differs from the standard one for the presence of those additional parameters that describe the employed random perturbation, and show that, similarly to the deterministic case, the stochastic reproduction number rules the asymptotic behavior of the solution. Aiming to make that random perturbation rigorous, we suggest an alternative approach based on a Wong-Zakai approximation argument thus arriving at a different stochastic model corresponding to the Stratonovich version of the Itô equation analyzed in Gray et al. Rather surprisingly, the asymptotic behavior of this alternative model turns out to be governed by the same reproduction number as the deterministic SIS equation. In other words, the random perturbation does not modify the threshold for extinction and persistence of the disease.
{{\copyright} 2022 Wiley Periodicals LLC.}Chase-escape percolation on the 2D square latticehttps://zbmath.org/1528.920402024-03-13T18:33:02.981707Z"Kumar, Aanjaneya"https://zbmath.org/authors/?q=ai:kumar.aanjaneya"Grassberger, Peter"https://zbmath.org/authors/?q=ai:grassberger.peter"Dhar, Deepak"https://zbmath.org/authors/?q=ai:dhar.deepakSummary: Chase-escape percolation is a variation of the standard epidemic spread models. In this model, each site can be in one of three states: unoccupied, occupied by a single prey, or occupied by a single predator. Prey particles spread to neighboring empty sites at rate \(p\), and predator particles spread only to neighboring sites occupied by prey particles at rate 1, killing the prey particle that existed at that site. It was found that the prey can survive forever with non-zero probability, if \(p > p_c\) with \(p_c < 1\). Earlier simulations showed that \(p_c\) is very close to 1/2. Using Monte Carlo simulations in \(D = 2\), we estimate the value of \(p_c\) to be \(0.49451 \pm 0.00001\) and the critical exponents are consistent with the undirected percolation universality class. We check that at \(p_c\), the correlation functions at large length scales are rotationally invariant. We define a discrete-time parallel-update version of the model, which brings out the relation between chase-escape and undirected bond percolation. We further show that for all \(p < p_c\), in \(D\) dimensions, the probability that the number of predators in the absorbing configuration is greater than \(s\) is bounded from below by \(\exp(-Kp^{-1} s^{1/D})\), where \(K\) is some \(p\)-independent constant. This is in contrast to the exponentially decaying cluster size distribution in the standard percolation theory. Even so, the scaling function for the cluster size distribution for \(p\) near \(p_c\) decays exponentially: the stretched exponential behavior dominates for \(s \gg s^\ast\), but \(s^\ast\) diverges near \(p_c\). We also study the problem starting from an initial condition with predator particles on all lattice points of the line \(y = 0\) and prey particles on the line \(y = 1\). In this case, for \(p_c < p < 1\), the center of mass of the fluctuating prey and predator fronts travel at the same speed. This speed is strictly smaller than the speed of an Eden front with the same value of \(p\), but with no predators. This is caused by the prey sites at the leading edge being eaten up by predators. The fluctuations of the front follow KPZ scaling both above and below the depinning transition at \(p = 1\).Higher order stochastically perturbed SIRS epidemic model with relapse and media impacthttps://zbmath.org/1528.920412024-03-13T18:33:02.981707Z"Rajasekar, S. P."https://zbmath.org/authors/?q=ai:rajasekar.s-p"Pitchaimani, M."https://zbmath.org/authors/?q=ai:pitchaimani.m"Zhu, Quanxin"https://zbmath.org/authors/?q=ai:zhu.quanxin(no abstract)A stochastic HIV/HTLV-I co-infection model incorporating the AIDS-related cancer cellshttps://zbmath.org/1528.920432024-03-13T18:33:02.981707Z"Yang, Hongbin"https://zbmath.org/authors/?q=ai:yang.hongbin"Li, Xiaoyue"https://zbmath.org/authors/?q=ai:li.xiaoyue"Zhang, Weipeng"https://zbmath.org/authors/?q=ai:zhang.weipengSummary: This paper proposes a stochastic HIV/HTLV-I co-infection model incorporating the AIDS-related cancer cells and investigates its dynamical behaviours. The main methods are stochastic Lyapunov analysis and the ergodic theory. The existence and uniqueness of the global positive solution is proved as well as the stochastic ultimate boundedness. A unique stationary distribution is yielded. Furthermore, the sufficient conditions for the extinction of diseases are given. Finally, some numerical simulations are carried out to support the theoretical results.Incremental nonlinear stability analysis of stochastic systems perturbed by Lévy noisehttps://zbmath.org/1528.931772024-03-13T18:33:02.981707Z"Han, SooJean"https://zbmath.org/authors/?q=ai:han.soojean"Chung, Soon-Jo"https://zbmath.org/authors/?q=ai:chung.soon-joSummary: We present a theoretical framework for characterizing incremental stability of nonlinear stochastic systems perturbed by either compound Poisson shot noise or finite-measure Lévy noise. For each noise type, we compare trajectories of the perturbed system with distinct noise sample paths against trajectories of the nominal, unperturbed system. We show that for a finite number of jumps arising from the noise process, the mean-squared error between the trajectories exponentially converge toward a bounded error ball across a finite interval of time under practical boundedness assumptions. The convergence rate for shot noise systems is the same as the exponentially stable nominal system, but with a tradeoff between the parameters of the shot noise process and the size of the error ball. The convergence rate and the error ball for the Lévy noise system are shown to be nearly direct sums of the respective quantities for the shot and white noise systems separately, a result which is analogous to the Lévy-Khintchine theorem. We demonstrate both empirical and analytical computation of the error ball using several numerical examples, and illustrate how varying the parameters of the system affect the tightness of the bound.
{{\copyright} 2022 John Wiley \& Sons Ltd.}Optimal finite-dimensional controller of the stochastic differential object's state by its output. I: Incomplete precise measurementshttps://zbmath.org/1528.932402024-03-13T18:33:02.981707Z"Rudenko, E. A."https://zbmath.org/authors/?q=ai:rudenko.e-aSummary: The well-known problem of synthesizing the optimal on the average and on given time interval of the inertial control law for a continuous stochastic object if only a part of its state variables are accurately measured is considered. Due to the practical unrealizability of its classical infinite-dimensional Stratonovich-Mortensen solution, it is proposed to limit ourselves to optimizing the structure of a finite-dimensional dynamic controller, whose order is chosen by the user. This finiteness allows using a truncated version of the a posteriori probability density that satisfies a deterministic partial differential integrodifferential equation. Using the Krotov extension principle, sufficient optimality conditions for the structural functions of the controller and the Lagrange-Pontryagin equation for finding their extremals are obtained. It is shown that in particular cases of the absence of measurements, complete measurements and taking into account only the values of incomplete measurements, the proposed controller turns out to be static (inertialess), and the relations for its synthesis coincide with the known ones. For a dynamic controller, algorithms for finding each of its structural functions are given.Geometric science of information. 6th international conference, GSI 2023, St. Malo, France, August 30 -- September 1, 2023. Proceedings. Part Ihttps://zbmath.org/1528.940032024-03-13T18:33:02.981707ZThe articles of this volume will be reviewed individually. For the preceding conference see [Zbl 1482.94007]. For Part II of the proceedings of the present conference see [Zbl 1528.53002].
Indexed articles:
\textit{Tumpach, Alice Barbora; Preston, Stephen C.}, Three methods to put a Riemannian metric on shape space, 3-11 [Zbl 07789178]
\textit{Maignant, Elodie; Trouvé, Alain; Pennec, Xavier}, Riemannian locally linear embedding with application to Kendall shape spaces, 12-20 [Zbl 07789179]
\textit{Pryymak, Lidiya; Suchan, Tim; Welker, Kathrin}, A product shape manifold approach for optimizing piecewise-smooth shapes, 21-30 [Zbl 07789180]
\textit{Tumpach, Alice Barbora}, On canonical parameterizations of 2D-shapes, 31-40 [Zbl 07789181]
\textit{Ciuclea, Ioana; Tumpach, Alice Barbora; Vizman, Cornelia}, Shape spaces of nonlinear flags, 41-50 [Zbl 07789182]
\textit{Bolelli, Maria Virginia; Citti, Giovanna; Sarti, Alessandro; Zucker, Steven}, A neurogeometric stereo model for individuation of 3D perceptual units, 53-62 [Zbl 07789183]
\textit{Pai, Gautam; Bellaard, Gijs; Smets, Bart M. N.; Duits, Remco}, Functional properties of PDE-based group equivariant convolutional neural networks, 63-72 [Zbl 07789184]
\textit{Vadgama, Sharvaree; Tomczak, Jakub M.; Bekkers, Erik}, Continuous Kendall shape variational autoencoders, 73-81 [Zbl 07789185]
\textit{Velasco-Forero, Santiago}, Can generalised divergences help for invariant neural networks?, 82-90 [Zbl 07789186]
\textit{Shewmake, Christian; Miolane, Nina; Olshausen, Bruno}, Group equivariant sparse coding, 91-101 [Zbl 07789187]
\textit{Broniatowski, Michel; Stummer, Wolfgang}, On a cornerstone of bare-simulation distance/divergence optimization, 105-116 [Zbl 07789188]
\textit{Girardin, Valérie; Regnault, Philippe}, Extensive entropy functionals and non-ergodic random walks, 117-124 [Zbl 07789189]
\textit{Boukeloua, Mohamed; Keziou, Amor}, Empirical likelihood with censored data, 125-135 [Zbl 07789190]
\textit{Baudry, Jean-Patrick; Broniatowski, Michel; Thommeret, Cyril}, Aggregated tests based on supremal divergence estimators for non-regular statistical models, 136-144 [Zbl 07789191]
\textit{Nielsen, Frank}, Quasi-arithmetic centers, quasi-arithmetic mixtures, and the Jensen-Shannon \(\nabla \)-divergences, 147-156 [Zbl 07789192]
\textit{Tanaka, Hisatoshi}, Geometry of parametric binary choice models, 157-166 [Zbl 07789193]
\textit{Tojo, Koichi; Yoshino, Taro}, A \(q\)-analogue of the family of Poincaré distributions on the upper half plane, 167-175 [Zbl 07789194]
\textit{Nielsen, Frank; Okamura, Kazuki}, On the \(f\)-divergences between hyperboloid and Poincaré distributions, 176-185 [Zbl 07789195]
\textit{Cheng, Kaiming; Zhang, Jun}, \( \lambda \)-deformed evidence lower bound (\( \lambda \)-ELBO) using Rényi and Tsallis divergence, 186-196 [Zbl 07789196]
\textit{Opozda, Barbara}, On the tangent bundles of statistical manifolds, 199-206 [Zbl 07789197]
\textit{Mama Assandje, Prosper Rosaire; Dongho, Joseph}, Geometric properties of beta distributions, 207-216 [Zbl 07789198]
\textit{Herguey, Mopeng; Dongho, Joseph}, KV cohomology group of some KV structures on \(\mathbb{R}^2\), 217-225 [Zbl 07789199]
\textit{Yoshioka, Masaki; Tanaka, Fuyuhiko}, Alpha-parallel priors on a one-sided truncated exponential family, 226-235 [Zbl 07789200]
\textit{Subrahamanian Moosath, K. S.; Mahesh, T. V.}, Conformal submersion with horizontal distribution and geodesics, 236-243 [Zbl 07789201]
\textit{Mainiero, Tom}, Higher information from families of measures, 247-257 [Zbl 07789202]
\textit{Sergeant-Perthuis, Grégoire}, A categorical approach to statistical mechanics, 258-267 [Zbl 07789203]
\textit{Perrone, Paolo}, Categorical information geometry, 268-277 [Zbl 07789204]
\textit{Chen, Stephanie; Vigneaux, Juan Pablo}, Categorical magnitude and entropy, 278-287 [Zbl 07789205]
\textit{Rioul, Olivier}, A historical perspective on Schützenberger-Pinsker inequalities, 291-306 [Zbl 07789206]
\textit{Florin, Franck}, On Fisher information matrix, array manifold geometry and time delay estimation, 307-317 [Zbl 07789207]
\textit{Meneghetti, Fábio C. C.; Miyamoto, Henrique K.; Costa, Sueli I. R.; Costa, Max H. M.}, Revisiting lattice tiling decomposition and dithered quantisation, 318-327 [Zbl 07789208]
\textit{Wolfer, Geoffrey; Watanabe, Shun}, Geometric reduction for identity testing of reversible Markov chains, 328-337 [Zbl 07789209]
\textit{Vigneaux, Juan Pablo}, On the entropy of rectifiable and stratified measures, 338-346 [Zbl 07789210]
\textit{Ulmer, Susanne; Van, Do Tran; Huckemann, Stephan F.}, Exploring uniform finite sample stickiness, 349-356 [Zbl 07789211]
\textit{Lammers, Lars; Van, Do Tran; Nye, Tom M. W.; Huckemann, Stephan F.}, Types of stickiness in BHV phylogenetic tree spaces and their degree, 357-365 [Zbl 07789212]
\textit{Calissano, Anna; Maignant, Elodie; Pennec, Xavier}, Towards quotient barycentric subspaces, 366-374 [Zbl 07789213]
\textit{Szwagier, Tom; Pennec, Xavier}, Rethinking the Riemannian logarithm on flag manifolds as an orthogonal alignment problem, 375-383 [Zbl 07789214]
\textit{Thanwerdas, Yann; Pennec, Xavier}, Characterization of invariant inner products, 384-391 [Zbl 07789215]
\textit{Lambert, Marc; Bonnabel, Silvère; Bach, Francis}, Variational Gaussian approximation of the Kushner optimal filter, 395-404 [Zbl 07789216]
\textit{Han, Andi; Mishra, Bamdev; Jawanpuria, Pratik; Gao, Junbin}, Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry, 405-415 [Zbl 07789217]
\textit{Minh, Hà Quang}, Fisher-Rao Riemannian geometry of equivalent Gaussian measures on Hilbert space, 416-425 [Zbl 07789218]
\textit{Da Costa, Nathaël; Mostajeran, Cyrus; Ortega, Juan-Pablo}, The Gaussian kernel on the circle and spaces that admit isometric embeddings of the circle, 426-435 [Zbl 07789219]
\textit{Said, Salem; Mostajeran, Cyrus}, Determinantal expressions of certain integrals on symmetric spaces, 436-443 [Zbl 07789220]
\textit{Chevallier, Emmanuel}, Projective Wishart distributions, 444-451 [Zbl 07789221]
\textit{Améndola, Carlos; Lee, Darrick; Meroni, Chiara}, Convex hulls of curves: volumes and signatures, 455-464 [Zbl 07789222]
\textit{Caravantes, Jorge; Diaz-Toca, Gema M.; Gonzalez-Vega, Laureano}, Avoiding the general position condition when computing the topology of a real algebraic plane curve defined implicitly, 465-473 [Zbl 07789223]
\textit{Gakkhar, Sita; Marcolli, Matilde}, Dynamical geometry and a persistence \(K\)-theory in noisy point clouds, 474-483 [Zbl 07789224]
\textit{Vermeylen, Charlotte; Olikier, Guillaume; Van Barel, Marc}, An approximate projection onto the tangent cone to the variety of third-order tensors of bounded tensor-train rank, 484-493 [Zbl 07789225]
\textit{Duarte, Eliana; Hollering, Benjamin; Wiesmann, Maximilian}, Toric fiber products in geometric modeling, 494-503 [Zbl 07789226]
\textit{Aniello, Paolo}, Twirled products and group-covariant symbols, 507-515 [Zbl 07789227]
\textit{Barron, Tatyana; Kazachek, Alexander}, Coherent states and entropy, 516-523 [Zbl 07789228]
\textit{Bieliavsky, Pierre; Dendoncker, Valentin}, A non-formal formula for the Rankin-Cohen deformation quantization, 524-532 [Zbl 07789229]
\textit{Bieliavsky, Pierre; Dendoncker, Valentin; Korvers, Stéphane}, Equivalence of invariant star-products: the ``retract'' method, 533-539 [Zbl 07789230]
\textit{Erzmann, David; Dittmer, Sören; Harms, Henrik; Maaß, Peter}, \texttt{DL4TO}: a deep learning library for sample-efficient topology optimization, 543-551 [Zbl 07789231]
\textit{Noren, Håkon}, Learning Hamiltonian systems with mono-implicit Runge-Kutta methods, 552-559 [Zbl 07789232]
\textit{Sutton, Oliver J.; Gorban, Alexander N.; Tyukin, Ivan Y.}, A geometric view on the role of nonlinear feature maps in few-shot learning, 560-568 [Zbl 07789233]
\textit{Offen, Christian; Ober-Blöbaum, Sina}, Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves, 569-579 [Zbl 07789234]
\textit{Huang, Qiao; Zambrini, Jean-Claude}, Gauge transformations in stochastic geometric mechanics, 583-591 [Zbl 07789235]
\textit{Bénéfice, Magalie; Arnaudon, Marc; Bonnefont, Michel}, Couplings of Brownian motions on \(\mathrm{SU}(2,\mathbb{C})\), 592-600 [Zbl 07789236]
\textit{Bhauryal, Neeraj}, A finite volume scheme for fractional conservation laws driven by Lévy noise, 601-609 [Zbl 07789237]
\textit{Chambolle, Antonin; Duval, Vincent; Machado, João Miguel}, The total variation-Wasserstein problem: a new derivation of the Euler-Lagrange equations, 610-619 [Zbl 07789238]Random sampling of signals concentrated on compact set in localized reproducing kernel subspace of \(L^p (\mathbb{R}^n)\)https://zbmath.org/1528.940172024-03-13T18:33:02.981707Z"Patel, Dhiraj"https://zbmath.org/authors/?q=ai:patel.dhiraj"Sivananthan, S."https://zbmath.org/authors/?q=ai:sivananthan.sivalingamSummary: The paper is devoted to studying the stability of random sampling in a localized reproducing kernel space. We show that if the sampling set on \(\Omega\) (compact) discretizes the integral norm of simple functions up to a given error, then the sampling set is stable for the set of functions concentrated on \(\Omega\). Moreover, we prove with an overwhelming probability that \(\mathcal{O}(\mu (\Omega )(\log \mu (\Omega ))^3)\) many random points uniformly distributed over \(\Omega\) yield a stable set of sampling for functions concentrated on \(\Omega\).Information propagation in stochastic networkshttps://zbmath.org/1528.940222024-03-13T18:33:02.981707Z"Juhász, Péter L."https://zbmath.org/authors/?q=ai:juhasz.peter-lSummary: In this paper, a network-based stochastic information propagation model is developed. The information flow is modeled by a probabilistic differential equation system. The numerical solution of these equations leads to the expected number of informed nodes as a function of time and reveals the relationship between the degrees of the nodes and their reception time. The validity of the model is justified by Monte Carlo network simulation through the analysis of information propagation in scale-free and Erdős-Rényi networks. It has been found that the developed model provides more accurate results compared to the widely used network-based SI mean-field model, especially in sparse networks.