Recent zbMATH articles in MSC 60https://zbmath.org/atom/cc/602023-01-20T17:58:23.823708ZWerkzeugThe Glauber dynamics for edge-colorings of treeshttps://zbmath.org/1500.050172023-01-20T17:58:23.823708Z"Delcourt, Michelle"https://zbmath.org/authors/?q=ai:delcourt.michelle"Heinrich, Marc"https://zbmath.org/authors/?q=ai:heinrich.marc"Perarnau, Guillem"https://zbmath.org/authors/?q=ai:perarnau.guillemSummary: Let \(T\) be a tree on \(n\) vertices and with maximum degree \(\Delta\). We show that for \(k \ge \Delta+1\) the Glauber dynamics for \(k\)-edge-colorings of \(T\) mixes in polynomial time in \(n\). The bound on the number of colors is best possible as the chain is not even ergodic for \(k \le \Delta\). Our proof uses a recursive decomposition of the tree into subtrees; we bound the relaxation time of the original tree in terms of the relaxation time of its subtrees using block dynamics and chain comparison techniques. Of independent interest, we also introduce a monotonicity result for Glauber dynamics that simplifies our proof.Some analytical properties of the matrix related to \(q\)-coloured Delannoy numbershttps://zbmath.org/1500.050372023-01-20T17:58:23.823708Z"Mu, Lili"https://zbmath.org/authors/?q=ai:mu.lili"Zheng, Sai-Nan"https://zbmath.org/authors/?q=ai:zheng.sainanSummary: The \(q\)-coloured Delannoy numbers \(D_{n,k}(q)\) count the number of lattice paths from \((0,\,0)\) to \((n,\,k)\) using steps \((0,\,1)\), \((1,\,0)\) and \((1,\,1)\), among which the \((1,\,1)\) steps are coloured with \(q\) colours. The focus of this paper is to study some analytical properties of the polynomial matrix \(D(q)=[d_{n,k}(q)]_{n,k\geq 0}=[D_{n-k,k}(q)]_{n,k\geq 0}\), such as the strong \(q\)-log-concavity of polynomial sequences located in a ray or a transversal line of \(D(q)\) and the \(q\)-total positivity of \(D(q)\). We show that the zeros of all row sums \(R_n(q)=\sum \nolimits_{k=0}^nd_{n,k}(q)\) are in \((-\infty,\, -1)\) and are dense in the corresponding semi-closed interval. We also prove that the zeros of all antidiagonal sums \(A_n(q)=\sum \nolimits_{k=0}^{\lfloor n/2 \rfloor}d_{n-k,k}(q)\) are in the interval \((-\infty,\, -1]\) and are dense there.Recurrence of the uniform infinite half-plane map via duality of resistanceshttps://zbmath.org/1500.050562023-01-20T17:58:23.823708Z"Budzinski, Thomas"https://zbmath.org/authors/?q=ai:budzinski.thomas"Lehéricy, Thomas"https://zbmath.org/authors/?q=ai:lehericy.thomasSummary: We study the simple random walk on the Uniform Infinite Half-Plane Map, which is the local limit of critical Boltzmann planar maps with a large and simple boundary. We prove that the simple random walk is recurrent, and that the resistance between the root and the boundary of the hull of radius \(r\) is at least of order \(\log r\). This resistance bound is expected to be sharp, and is better than those following from previous proofs of recurrence for nonbounded-degree planar maps models. Our main tools are the self-duality of uniform planar maps, a classical lemma about duality of resistances and some peeling estimates. The proof shares some ideas with Russo-Seymour-Welsh theory in percolation.Optimal and algorithmic norm regularization of random matriceshttps://zbmath.org/1500.150322023-01-20T17:58:23.823708Z"Jain, Vishesh"https://zbmath.org/authors/?q=ai:jain.vishesh"Sah, Ashwin"https://zbmath.org/authors/?q=ai:sah.ashwin"Sawhney, Mehtaab"https://zbmath.org/authors/?q=ai:sawhney.mehtaab-sAuthors' abstract: Let \(A\) be an \(n\times n\) random matrix whose entries are i.i.d. with mean \(0\) and variance \(1\). We present a deterministic polynomial time algorithm which, with probability at least \(1-2\exp(-\Omega(\epsilon n))\) in the choice of \(A\), finds an \(\epsilon n\times\epsilon n\) sub-matrix such that zeroing it out results in \(\widetilde{A}\) with
\[
\Vert\widetilde{A}\Vert=O(\sqrt{n/\epsilon}).
\]
Our result is optimal up to a constant factor and improves previous results of \textit{E. Rebrova} and \textit{R. Vershynin} [Adv. Math. 324, 40--83 (2018; Zbl 1380.60016)], and \textit{E. Rebrova} [J. Theor. Probab. 33, No. 3, 1768--1790 (2020; Zbl 1444.60014)]. We also prove an analogous result for \(A\) a symmetric \(n\times n\) random matrix whose upper-diagonal entries are i.i.d. with mean \(0\) and variance \(1\).
Reviewer: Sho Matsumoto (Kagoshima)Singularity of energy measures on a class of inhomogeneous Sierpinski gasketshttps://zbmath.org/1500.280072023-01-20T17:58:23.823708Z"Hino, Masanori"https://zbmath.org/authors/?q=ai:hino.masanori"Yasui, Madoka"https://zbmath.org/authors/?q=ai:yasui.madokaSummary: We study energy measures of canonical Dirichlet forms on inhomogeneous Sierpinski gaskets. We prove that the energy measures and suitable reference measures are mutually singular under mild assumptions.
For the entire collection see [Zbl 1493.11005].Qualitative analysis of an HIV/AIDS model with treatment and nonlinear perturbationhttps://zbmath.org/1500.340392023-01-20T17:58:23.823708Z"Gao, Miaomiao"https://zbmath.org/authors/?q=ai:gao.miaomiao"Jiang, Daqing"https://zbmath.org/authors/?q=ai:jiang.daqing"Hayat, Tasawar"https://zbmath.org/authors/?q=ai:hayat.tasawarSummary: In this paper, we consider a high-dimensional stochastic HIV/AIDS model that incorporates both multiple stages treatment and higher order perturbation. Firstly, we establish sufficient criteria for the existence of a unique ergodic stationary distribution by making use of stochastic Lyapunov analysis method. Stationary distribution shows that the disease will be persistent in the long term. Then, conditions for extinction of the disease are obtained. Theoretical analysis indicates that large noise intensity can suppress the prevalence of HIV/AIDS epidemic. Finally, we provide some numerical simulations to illustrate the analytical results.Dynamics of a stochastic HIV/AIDS model with treatment under regime switchinghttps://zbmath.org/1500.340402023-01-20T17:58:23.823708Z"Gao, Miaomiao"https://zbmath.org/authors/?q=ai:gao.miaomiao"Jiang, Daqing"https://zbmath.org/authors/?q=ai:jiang.daqing"Hayat, Tasawar"https://zbmath.org/authors/?q=ai:hayat.tasawar"Alsaedi, Ahmed"https://zbmath.org/authors/?q=ai:alsaedi.ahmed"Ahmad, Bashir"https://zbmath.org/authors/?q=ai:ahmad.bashir.1|ahmad.bashir.2Summary: This paper focuses on the spread dynamics of an HIV/AIDS model with multiple stages of infection and treatment, which is disturbed by both white noise and telegraph noise. Switching between different environmental states is governed by Markov chain. Firstly, we prove the existence and uniqueness of the global positive solution. Then we investigate the existence of a unique ergodic stationary distribution by constructing suitable Lyapunov functions with regime switching. Furthermore, sufficient conditions for extinction of the disease are derived. The conditions presented for the existence of stationary distribution improve and generalize the previous results. Finally, numerical examples are given to illustrate our theoretical results.Besicovitch almost periodic solutions to stochastic dynamic equations with delayshttps://zbmath.org/1500.340702023-01-20T17:58:23.823708Z"Li, Yongkun"https://zbmath.org/authors/?q=ai:li.yongkun|li.yongkun.1"Huang, Xiaoli"https://zbmath.org/authors/?q=ai:huang.xiaoliSummary: In order to unify the study of Besicovitch almost periodic solutions of continuous time and discrete-time stochastic differential equations, we first propose concepts of Besicovitch almost periodic stochastic processes in \(p\)-th mean and of Besicovitch almost periodic stochastic processes in distribution on time scales, and reveal the relationship between the two random processes. Then, taking a class of stochastic Clifford-valued neural networks with time-varying delays on time scales as an example of stochastic dynamic equations with delays, we establish the existence and stability of Besicovitch almost periodic solutions in distribution for this class of networks by using Banach's fixed point theorem, time scale calculus theory and inequality techniques.Heat kernels of non-local Schrödinger operators with Kato potentialshttps://zbmath.org/1500.350042023-01-20T17:58:23.823708Z"Grzywny, Tomasz"https://zbmath.org/authors/?q=ai:grzywny.tomasz"Kaleta, Kamil"https://zbmath.org/authors/?q=ai:kaleta.kamil"Sztonyk, Paweł"https://zbmath.org/authors/?q=ai:sztonyk.pawelSummary: We study heat kernels of Schrödinger operators whose kinetic terms are non-local operators built for sufficiently regular symmetric Lévy measures with radial decreasing profiles and potentials belong to Kato class. Our setting is fairly general and novel -- it allows us to treat both heavy- and light-tailed Lévy measures in a joint framework. We establish a certain relative-Kato bound for the corresponding semigroups and potentials. This enables us to apply a general perturbation technique to construct the heat kernels and give sharp estimates of them. Assuming that the Lévy measure and the potential satisfy a little stronger conditions, we additionally obtain the regularity of the heat kernels. Finally, we discuss the applications to the smoothening properties of the corresponding semigroups. Our results cover many important examples of non-local operators, including \textit{fractional} and \textit{quasi-relativistic} Schrödinger operators.Quantitative homogenization of interacting particle systemshttps://zbmath.org/1500.350212023-01-20T17:58:23.823708Z"Giunti, Arianna"https://zbmath.org/authors/?q=ai:giunti.arianna"Gu, Chenlin"https://zbmath.org/authors/?q=ai:gu.chenlin"Mourrat, Jean-Christophe"https://zbmath.org/authors/?q=ai:mourrat.jean-christopheSummary: For a class of interacting particle systems in continuous space, we show that finite-volume approximations of the bulk diffusion matrix converge at an algebraic rate. The models we consider are reversible with respect to the Poisson measures with constant density, and are of nongradient type. Our approach is inspired by recent progress in the quantitative homogenization of elliptic equations. Along the way, we develop suitable modifications of the Caccioppoli and multiscale Poincaré inequalities, which are of independent interest.Symptotic decomposition of solutions to parabolic equations with a random microstructurehttps://zbmath.org/1500.350232023-01-20T17:58:23.823708Z"Kleptsyna, Marina"https://zbmath.org/authors/?q=ai:kleptsyna.marina-l"Piatnitski, Andrey"https://zbmath.org/authors/?q=ai:piatnitski.andrey-l"Popier, Alexandre"https://zbmath.org/authors/?q=ai:popier.alexandreSummary: We consider a Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variables and random stationary ergodic in time. As was proved in [\textit{V. V. Zhikov} et al., Tr. Mosk. Mat. O.-va 45, 182--236 (1982; Zbl 0531.35041)] and [the first and second author, GAKUTO Int. Ser., Math. Sci. Appl. 9, 241--255 (1995; Zbl 0892.35019)] in this case the homogenized operator is deterministic.
We obtain the leading terms of the asymptotic expansion of the solution, these terms being deterministic functions, and show that a properly renormalized difference between the solution and the said leading terms converges to a solution of some SPDE.Controller and asymptotic autonomy of random attractors for stochastic p-Laplace lattice equationshttps://zbmath.org/1500.350532023-01-20T17:58:23.823708Z"Song, Li"https://zbmath.org/authors/?q=ai:song.li"Li, Yangrong"https://zbmath.org/authors/?q=ai:li.yangrong"Wang, Fengling"https://zbmath.org/authors/?q=ai:wang.fenglingSummary: A non-autonomous random dynamical system is called to be controllable if there is a pullback random attractor (PRA) such that each fibre of the PRA converges upper semi-continuously to a nonempty compact set (called a controller) as the time-parameter goes to minus infinity, while the PRA is called to be asymptotically autonomous if there is a random attractor for another (autonomous) random dynamical system as a controller. We establish the criteria for ensuring the existence of the minimal controller and the asymptotic autonomy of a PRA respectively. The abstract results are illustrated in possibly non-autonomous stochastic p-Laplace lattice equations with tempered convergent external forces.Random attractors of supercritical stochastic wave equationshttps://zbmath.org/1500.350552023-01-20T17:58:23.823708Z"Wang, Bixiang"https://zbmath.org/authors/?q=ai:wang.bixiangSummary: This paper is concerned with the well-posedness and the asymptotic behavior of solutions of the non-autonomous stochastic wave equations with supercritical nonlinearity driven by additive white noise defined in bounded domains. By using the Strichartz inequalities, we first prove the existence and uniqueness of solutions of the stochastic equation with the homogeneous Dirichlet boundary condition. We then show the asymptotic compactness of solutions and hence the existence of tempered pullback random attractors by a decomposition method.Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and Stein kernelshttps://zbmath.org/1500.350652023-01-20T17:58:23.823708Z"Fathi, Max"https://zbmath.org/authors/?q=ai:fathi.max"Mikulincer, Dan"https://zbmath.org/authors/?q=ai:mikulincer.danSummary: We investigate stability of invariant measures of diffusion processes with respect to \(L^p\) distances on the coefficients, under an assumption of log concavity. The method is a variant of a technique introduced by Crippa and De Lellis to study transport equations. As an application, we prove a partial extension of an inequality of Ledoux, Nourdin and Peccati relating transport distances and Stein discrepancies to a non-Gaussian setting via the moment map construction of Stein kernels.Optimal regularity in time and space for stochastic porous medium equationshttps://zbmath.org/1500.350722023-01-20T17:58:23.823708Z"Bruno, Stefano"https://zbmath.org/authors/?q=ai:bruno.stefano"Gess, Benjamin"https://zbmath.org/authors/?q=ai:gess.benjamin"Weber, Hendrik"https://zbmath.org/authors/?q=ai:weber.hendrikSummary: We prove optimal regularity estimates in Sobolev spaces in time and space for solutions to stochastic porous medium equations. The noise term considered here is multiplicative, white in time and coloured in space. The coefficients are assumed to be Hölder continuous, and the cases of smooth coefficients of, at most, linear growth as well as \(\sqrt{u}\) are covered by our assumptions. The regularity obtained is consistent with the optimal regularity derived for the deterministic porous medium equation in [\textit{B. Gess}, J. Eur. Math. Soc. (JEMS) 23, No. 2, 425--465 (2021; Zbl 1461.35133); \textit{B. Gess} et al., Anal. PDE 13, No. 8, 2441--2480 (2020; Zbl 1459.35254)] and the presence of the temporal white noise. The proof relies on a significant adaptation of velocity averaging techniques from their usual \(L^1\) context to the natural \(L^2\) setting of the stochastic case. We introduce a new mixed kinetic/mild representation of solutions to quasilinear SPDE and use \(L^2\) based a priori bounds to treat the stochastic term.Optimal sizing of the sediment replenishment capacity based on robust ergodic control of subordinator-driven dynamicshttps://zbmath.org/1500.350932023-01-20T17:58:23.823708Z"Yoshioka, Hidekazu"https://zbmath.org/authors/?q=ai:yoshioka.hidekazu"Tsujimura, Motoh"https://zbmath.org/authors/?q=ai:tsujimura.motohSummary: A countermeasure against sediment depletion in rivers is sand replenishment. The cost-efficient sizing of its capacity is critical for sustainable environmental management. However, only temporally discrete replenishment is possible, and sediment transport is intermittent and uncertain. In this paper, we approach this new design problem using a robust ergodic control. First, a stochastic differential equation driven by a (tempered) stable subordinator describes the sediment storage dynamics. Sand replenishment is considered an impulse control, wherein observation/replenishment chances arrive at Poisson jump times. The subordinator is ambiguous since the decision-maker does not precisely know its Lévy measure. Using a dynamic risk measure that balances replenishment cost, disutility triggered by sediment depletion, and penalization of ambiguity, we derived the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation to determine the optimal storage capacity. We obtained that the HJBI equation has an analytical solution when a stable subordinator is used. Furthermore, we prove it admits a unique continuous viscosity solution and that it is optimal and applied the proposed model to real data.Quantitative homogenization in a balanced random environmenthttps://zbmath.org/1500.351172023-01-20T17:58:23.823708Z"Guo, Xiaoqin"https://zbmath.org/authors/?q=ai:guo.xiaoqin"Peterson, Jonathon"https://zbmath.org/authors/?q=ai:peterson.jonathon"Tran, Hung V."https://zbmath.org/authors/?q=ai:tran.hung-vinhSummary: We consider discrete non-divergence form difference operators in a random environment and the corresponding process -- the random walk in a balanced random environment in \({\mathbb{Z}^d}\) with a finite range of dependence. We first quantify the ergodicity of the \textit{environment from the point of view of the particle}. As a consequence, we quantify the quenched central limit theorem of the random walk with an algebraic rate. Furthermore, we prove an algebraic rate of convergence for the homogenization of the Dirichlet problems for both elliptic and parabolic non-divergence form difference operators.Continuity problem for singular BSDE with random terminal timehttps://zbmath.org/1500.351732023-01-20T17:58:23.823708Z"Samuel, Sharoy Augustine"https://zbmath.org/authors/?q=ai:samuel.sharoy-augustine"Popier, Alexandre"https://zbmath.org/authors/?q=ai:popier.alexandre"Sezer, Ali Devin"https://zbmath.org/authors/?q=ai:sezer.ali-devinSummary: We study a class of non-linear Backward stochastic differential equations (BSDE) with a superlinear driver process \(f\) adapted to a filtration \(\mathbb{F}\) and over a random time interval \textlbrackdbl\(0, S\)\textrbrackdbl\,where \(S\) is a stopping time of \(\mathbb{F}\). The terminal condition \(\xi\) is allowed to take the value \(+\infty\), i.e., singular. We call a stopping time \(S\) solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value \(\infty\) at terminal time \(S\). Our goal is to show existence of solutions to the BSDE for a range of singular terminal values under the assumption that \(S\) is solvable. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., it is continuous at time \(S\) and attains the terminal value with probability 1. We consider three types of terminal values: 1) Markovian: i.e., \(\xi\) is of the form \(\xi=g (\Xi_S)\) where \(\Xi\) is a continuous Markovian diffusion process, \(S\) is a hitting time of \(\Xi\) and \(g\) is a deterministic function 2) terminal conditions of the form \(\xi_1=\infty \cdot \mathbf{1}_{\{\tau \leq S\}}\) and 3) \(\xi_2=\infty \cdot \mathbf{1}_{\{\tau>S\}}\) where \(\tau\) is another stopping time. For general \(\xi\) we prove that minimal supersolution has a limit at time \(S\) provided that \(\mathbb{F}\) is left continuous at time \(S\). Finally, we discuss the implications of our results about Markovian terminal conditions to the solution of non-linear elliptic PDE with singular boundary conditions.Heat kernel bounds for parabolic equations with singular (form-bounded) vector fieldshttps://zbmath.org/1500.351972023-01-20T17:58:23.823708Z"Kinzebulatov, Damir"https://zbmath.org/authors/?q=ai:kinzebulatov.damir"Semënov, Yuliy A."https://zbmath.org/authors/?q=ai:semenov.yuliy-aSummary: We consider Kolmogorov operator \(-\nabla \cdot a \cdot \nabla + b \cdot \nabla\) with measurable uniformly elliptic matrix \(a\) and prove Gaussian lower and upper bounds on its heat kernel under minimal assumptions on the vector field \(b\) and its divergence \(\text{div\,}b\). More precisely, we prove: (1) Gaussian lower bound, provided that \(\text{div\,}b \ge 0\), and \(b\) is in the class of form-bounded vector fields (containing e.g. the class \(L^d\), the weak \(L^d\) class, as well as some vector fields that are not even in \(L_{\text{loc}}^{2+\varepsilon }, \varepsilon >0)\); in these assumptions, the Gaussian upper bound is in general invalid; (2) Gaussian upper bound, provided that \(b\) is form-bounded, and the positive part of \(\text{div\,}b\) is in the Kato class; in these assumptions, the Gaussian lower bound is in general invalid; (3) Gaussian upper and lower bounds, provided that \(b\) is form-bounded, \( \text{div\,}b\) is in the Kato class; (4) A priori Gaussian upper and lower bounds, provided that \(b\) is in a large class containing the class of form-bounded vector fields, \( \text{div\,}b\) is in the Kato class.Multiple Riemann wave solutions of the general form of quasilinear hyperbolic systemshttps://zbmath.org/1500.352362023-01-20T17:58:23.823708Z"Grundland, A. M."https://zbmath.org/authors/?q=ai:grundland.alfred-michel"de Lucas, J."https://zbmath.org/authors/?q=ai:de-lucas.javierThe authors consider the quasi-linear first-order system of \(q\) equations in \(p\) independent variables \(\sum_{i=1}^{p}A^{i}(u)\frac{\partial u}{\partial x_{i}}=0\), \(i=1\ldots p\), where \(u=(u^{1},\ldots,u^{q})\in U\subset \mathbb{R}^{q} \) and \(A^{1},\ldots ,A^{p}\) are \(q\times q\) matrices. They define a Riemann wave solution to this system defined in a neighborhood of \(x=0\) as a solution to the equation \(u=f(r(x,u))\), where the functions \(f:\mathbb{R} \rightarrow \mathbb{R}^{q}\) and \(r:\mathbb{R}^{p+q}\rightarrow \mathbb{R}\) are given by \(r(x,u)=\sum_{i=1}^{p}\lambda _{i}(u)x^{i}\), the solution \(u\) satisfying, through the implicit function theorem and in a neighborhood of \(x=0\) : \(\frac{\partial u^{\alpha }}{\partial x^{i}}(x)=\phi (x)^{-1}\lambda _{i}(u(x))\frac{df^{\alpha }}{dr}(r(x,u(x)))\), \(i=1,\ldots ,p\), \(\alpha =1,\ldots ,q\), where \(\phi (x)=1-\sum_{\alpha =1}^{q}\frac{\partial r}{ \partial u^{\alpha }}(x,u(x))dfdr(r(x;u(x)))\). The function \(r\) is a Riemann invariant associated to the non-zero wave vector \(\lambda =(\lambda _{1},\ldots ,\lambda _{p}):\mathbb{R}^{p}\rightarrow \mathbb{R}^{q}\), satisfying \(ker(\sum_{i=1}^{p}\lambda _{i}(u)A^{i}(u))\neq 0\). The authors first analyze the properties of the quasi-linear hyperbolic system \( \sum_{i=1}^{p}\sum_{\beta =1}^{q}A_{\beta }^{i\alpha }(x,u)u_{i}^{\beta }\), \( \alpha =1,\ldots ,q\), of its integral elements and of its \(k\)-wave solutions. They establish conditions which lead to the existence of a Riemann \(k\)-wave solution. The paper ends with the analysis of 2D examples.
Reviewer: Alain Brillard (Riedisheim)Dispersionless version of the constrained Toda hierarchy and symmetric radial Löwner equationhttps://zbmath.org/1500.352652023-01-20T17:58:23.823708Z"Takebe, Takashi"https://zbmath.org/authors/?q=ai:takebe.takashi"Zabrodin, Anton"https://zbmath.org/authors/?q=ai:zabrodin.anton-vSummary: We study the dispersionless version of the recently introduced constrained Toda hierarchy. Like the Toda lattice itself, it admits three equivalent formulations: the formulation in terms of Lax equations, the formulation of the Zakharov-Shabat type and the formulation through the generating equation for the dispersionless limit of logarithm of the tau-function. We show that the dispersionless constrained Toda hierarchy describes conformal maps of reflection-symmetric planar domains to the exterior of the unit disc. We also find finite-dimensional reductions of the hierarchy and show that they are characterized by a differential equation of the Löwner type which we call the symmetric radial Löwner equation. It is also shown that solutions to the symmetric radial Löwner equation are conformal maps of the exterior of the unit circle with two symmetric slits to the exterior of the unit circle.Feeling boundary by Brownian motion in a ballhttps://zbmath.org/1500.352782023-01-20T17:58:23.823708Z"Serafin, G."https://zbmath.org/authors/?q=ai:serafin.grzegorzSummary: We establish short-time asymptotics with rates of convergence for the Laplace Dirichlet heat kernel in a ball. So far, such results were only known in simple cases where explicit formulae are available, i.e., for sets as half-line, interval and their products. Presented asymptotics may be considered as a complement or a generalization of the famous ``principle of not feeling the boundary'' in case of a ball. Following the metaphor, the principle reveals when the process does not feel the boundary, while we describe what happens when it starts feeling the boundary.Power mixture forward performance processeshttps://zbmath.org/1500.352802023-01-20T17:58:23.823708Z"Avanesyan, Levon"https://zbmath.org/authors/?q=ai:avanesyan.levon"Sircar, Ronnie"https://zbmath.org/authors/?q=ai:sircar.ronnieSummary: We consider the forward investment problem in market models where the stock prices are continuous semimartingales adapted to a Brownian filtration. We construct a broad class of forward performance processes with initial conditions of power mixture type, \(u(x) = \int_{\mathbb{I}} \frac{x^{1-\gamma}}{1-\gamma}\nu(d\gamma)\). We proceed to define and fully characterize two-power mixture forward performance processes with constant risk aversion coefficients in the interval \((0, 1)\), and derive properties of two-power mixture forward performance processes when the risk aversion coefficients are continuous stochastic processes. Finally, we discuss the problem of managing an investment pool of two investors, whose respective preferences evolve as power forward performance processes.Invariant measure for neutral stochastic functional differential equations with non-Lipschitz coefficientshttps://zbmath.org/1500.352892023-01-20T17:58:23.823708Z"Stanzhytsky, Andriy"https://zbmath.org/authors/?q=ai:stanzhytsky.andriy"Misiats, Oleksandr"https://zbmath.org/authors/?q=ai:misiats.oleksandr"Stanzhytskyi, Oleksandr"https://zbmath.org/authors/?q=ai:stanzhytskyi.oleksandr-mSummary: In this work we study the long time behavior of nonlinear stochastic functional-differential equations of neutral type in Hilbert spaces with non-Lipschitz nonlinearities. We establish the existence of invariant measures in the shift spaces for such equations. Our approach is based on Krylov-Bogoliubov theorem on the tightness of the family of measures.The continuity, regularity and polynomial stability of mild solutions for stochastic 2D-Stokes equations with unbounded delay driven by tempered fractional Gaussian noisehttps://zbmath.org/1500.353182023-01-20T17:58:23.823708Z"Liu, Yarong"https://zbmath.org/authors/?q=ai:liu.yarong"Wang, Yejuan"https://zbmath.org/authors/?q=ai:wang.yejuan"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomasThe authors consider the following stochastic 2D-Stokes equation with unbounded delay:
\[
\begin{aligned}
\mathrm{d} u(t) = \Delta u(t) \mathrm{d} t+ \delta u(t) \mathrm{d}t + \nabla p(t) \mathrm{d} t+ F(t, u_t)\mathrm{d} t + G(t,u_t) \mathrm{d} B^{\sigma,\lambda}(t) \quad & \text{ in }\mathbb{R}^2,\ t>0,\\
\nabla \cdot u = 0 \quad & \text{ in }\mathbb{R}^2,\ t>0,\\
u(t,x)=\varphi(t,x) \quad & \text{ in }\mathbb{R}^2,\ t \leq 0.
\end{aligned}
\]
The notation \(u_t\) stands for the function \(u_t(s)=u(t+s)\) defined for \(s\in (-\infty,0)\), which is why the equation displays unbounded delay; in other terms, the dynamics at time \(t\) depends on all the previous history of the solution on \((-\infty,t)\).
Instead \(B^{\sigma,\lambda}\) denotes Tempered Fractional Brownian Motion (TFBM), a centered Gaussian process which can be defined by the stochastic integral representation
\[
B^{\sigma,\lambda}(t)=\int_{-\infty}^{+\infty} \big[ e^{-\lambda(t-x)_+}(t-x)_{+}^{-\sigma} - e^{-\lambda (-x)_+} (-x)_+^{-\sigma} \big] \mathrm{d} B(x)
\]
for parameters \(\sigma\in (-1/2,0]\) and \(\lambda\geq 0\). For \(\lambda=0\), TFBM reduces to the more classical FBM with Hurst parameter \(H=1/2-\sigma\); the tempered parameter \(\lambda\) controls the deviation of \(B^{\sigma,\lambda}\) from a FBM power law spectrum at low frequencies. One of the main difficulties in solving equations driven by \(B^{\sigma,\lambda}\) is that, for \(\sigma\neq 0\), it is not a semimartingale nor a Markov process, thus Itô calculus is not available.
In this paper the authors first develop useful estimates for stochastic integrals w.r.t. \(B^{\sigma,\lambda}\) and then establish global existence and uniqueness of mild solutions for the above equation, in suitable function spaces and under standard Lipschitz and linear growth assumptions on \(F\), \(G\); here for \(\lambda\geq 0\) and \(\sigma\in (-1/2,0)\) (correspondingly \(H\in (1/2,1)\)). They then prove continuity of the solution map \((\sigma,\lambda)\mapsto u^{\sigma,\lambda}\), firstly as \(\lambda\to 0\) and then as \(\sigma_1\to\sigma_2\in (-1/2,0)\), at least on bounded time intervals. Time regularity of solutions is also obtained.
Finally, under more specific assumptions on the coefficients \(F\), \(G\), polynomial and exponential stability as \(t\to\infty\) are shown, in the sense e.g. of bounds of the form
\[
\sup_{r\in [0,\infty)}r^{\xi} \mathbb{E}[ \| u(r)\|_{H^\gamma}^p] <+\infty
\]
where \(\xi\in (0,1)\), \(\mathbb{E}\) denotes expectation and \(p\geq 2\).
Reviewer: Lucio Galeati (Lausanne)Entropy dimension for deterministic walks in random scenerieshttps://zbmath.org/1500.370052023-01-20T17:58:23.823708Z"Dou, Dou"https://zbmath.org/authors/?q=ai:dou.dou"Park, Kyewon Koh"https://zbmath.org/authors/?q=ai:park.kyewon-kohSummary: Entropy dimension is an entropy-type quantity which takes values in \([0,1]\) and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.Local limit theorem in deterministic systemshttps://zbmath.org/1500.370082023-01-20T17:58:23.823708Z"Kosloff, Zemer"https://zbmath.org/authors/?q=ai:kosloff.zemer"Volny, Dalibor"https://zbmath.org/authors/?q=ai:volny.daliborThe authors claim and prove a local limit theorem for deterministic systems.
The existence of functions satisfying the central limit theorem or the local central limit theorem has been considered by many authors, e.g., [\textit{G. Maruyama}, Proc. 3rd Japan-USSR Symp. Probab. Theory, Taschkent 1975, Lect. Notes Math. 550, 375-378 (1976; Zbl 0375.60045); \textit{R. Burton} and \textit{M. Denker}, Trans. Am. Math. Soc. 302, 715--726 (1987; Zbl 0628.60030)].
The present work is mostly influenced by the paper [\textit{D. Volný}, Trans. Am. Math. Soc. 351, No. 8, 3351--3371 (1999; Zbl 0939.37006)], where for every aperiodic dynamical system, a function satisfying the invariance principle and the almost sure invariance principle was constructed.
In the dynamical systems setting, it is in general a nontrivial problem to determine whether a function which satisfies the central limit theorem also satisfies the local central limit theorem. In fact, even in the nicer setting of chaotic (piecewise) smooth dynamical systems a local central limit theorem is usually proved under more stringent spectral conditions, see, e.g., the works [\textit{J. Aaronson} and \textit{M. Denker}, Stoch. Dyn. 1, No. 2, 193--237 (2001; Zbl 1039.37002); \textit{Y. Guivarc'h} and \textit{J. Hardy}, Ann. Inst. Henri Poincaré, Probab. Stat. 24, No. 1, 73--98 (1988; Zbl 0649.60041); \textit{J. Rousseau-Egele}, Ann. Probab. 11, 772--788 (1983; Zbl 0518.60033)]. But the methods used therein to prove the central limit theorem rely on non-spectral tools that are not optimal to obtain the local limit theorem. Moreover, the resulting partial sum function process takes countless values.
All these technical problems are solved by the theorem proved by the authors in this article.
Theorem. For every ergodic aperiodic and measure-preserving dynamical system \((X,B,m,T)\) there exists a square integrable function \(f:X\rightarrow \mathbb{Z}\) with \(\int f dm=0\) which satisfies the lattice local central limit theorem.
To construct functions satisfying the local central limit theorem the authors refer to [\textit{J. C. Kieffer}, Ann. Probab. 8, 131--141 (1980; Zbl 0426.60036)], where a new version of the stochastic coding theorem was given. More precisely, it was shown that in any ergodic aperiodic dynamical system it is possible to realize each independent triangular array by taking a finite number of values.
Reviewer: Tatiana Mamedova (Saransk)High-low temperature dualities for the classical \(\beta \)-ensembleshttps://zbmath.org/1500.370112023-01-20T17:58:23.823708Z"Forrester, Peter J."https://zbmath.org/authors/?q=ai:forrester.peter-jSummary: The loop equations for the \(\beta \)-ensembles are conventionally solved in terms of a \(1/N\) expansion. We observe that it is also possible to fix \(N\) and expand in inverse powers of \(\beta \). At leading order, for the one-point function \(W_1(x)\) corresponding to the average of the linear statistic \(A= \sum_{j = 1}^N 1/(x- \lambda_j)\) and after specialising to the classical weights, this reclaims well known results of Stieltjes relating the zeros of the classical polynomials to the minimum energy configuration of certain log-gas potential energies. Moreover, it is observed that the differential equations satisfied by \(W_1(x)\) in the case of classical weights -- which are particular Riccati equations -- are simply related to the differential equations satisfied by \(W_1(x)\) in the high temperature scaled limit \(\beta=2\alpha/N\) (\(\alpha\) fixed, \(N\to\infty\)), implying a certain high-low temperature duality. A generalisation of this duality, valid without any limiting procedure, is shown to hold for \(W_1(x)\) and all its higher point analogues in the classical \(\beta \)-ensembles.tgEDMD: approximation of the Kolmogorov operator in tensor train formathttps://zbmath.org/1500.370462023-01-20T17:58:23.823708Z"Lücke, Marvin"https://zbmath.org/authors/?q=ai:lucke.marvin"Nüske, Feliks"https://zbmath.org/authors/?q=ai:nuske.feliksSummary: Extracting information about dynamical systems from models learned off simulation data has become an increasingly important research topic in the natural and engineering sciences. Modeling the Koopman operator semigroup has played a central role in this context. As the approximation quality of any such model critically depends on the basis set, recent work has focused on deriving data-efficient representations of the Koopman operator in low-rank tensor formats, enabling the use of powerful model classes while avoiding over-fitting. On the other hand, detailed information about the system at hand can be extracted from models for the infinitesimal generator, also called Kolmogorov backward operator for stochastic differential equations. In this work, we present a data-driven method to efficiently approximate the generator using the tensor train (TT) format. The centerpiece of the method is a TT representation of the tensor of generator evaluations at all data sites. We analyze consistency and complexity of the method, present extensions to practically relevant settings, and demonstrate its applicability to benchmark numerical examples.Percolation of three fluids on a honeycomb latticehttps://zbmath.org/1500.420142023-01-20T17:58:23.823708Z"Novikov, Ivan"https://zbmath.org/authors/?q=ai:novikov.ivan-sergeevichSummary: In this paper, we consider a generalization of percolation: percolation of three related fluids on a honeycomb lattice. Izyurov and Magazinov (personal communication) proved that percolations of distinct fluids between opposite sides on a fixed hexagon become mutually independent as the lattice step tends to 0. This paper exposes this proof in details (with minor simplifications) for nonspecialists. In addition, we state a few related conjectures based on numerical experiments.Cameron-Storvick theorem associated with Gaussian paths on function spacehttps://zbmath.org/1500.460362023-01-20T17:58:23.823708Z"Choi, Jae Gil"https://zbmath.org/authors/?q=ai:choi.jae-gilAuthor's abstract: The purpose of this paper is to provide a more general Cameron-Storvick theorem for the generalized analytic Feynman integral associated with Gaussian process \(\mathcal{Z}_k\) on a very general Wiener space \(C_{a, b}[0, T]\). The general Wiener space \(C_{a, b}[0, T]\) can be considered as the set of all continuous sample paths of the generalized Brownian motion process determined by continuous functions \(a(t)\) and \(b(t)\) on \([0, T]\). As an interesting application, we apply this theorem to evaluate the generalized analytic Feynman integral of certain monomials in terms of Paley-Wiener-Zygmund stochastic integrals.
Reviewer: Denis R. Bell (Jacksonville)Transfer operators, induced probability spaces, and random walk modelshttps://zbmath.org/1500.471232023-01-20T17:58:23.823708Z"Jorgensen, P."https://zbmath.org/authors/?q=ai:jorgensen.palle-e-t"Tian, F."https://zbmath.org/authors/?q=ai:tian.fengSummary: We study a family of discrete-time random-walk models. The starting point is a fixed generalized transfer operator \(R\) subject to a set of axioms, and a given endomorphism in a compact Hausdorff space \(X\). Our setup includes a host of models from applied dynamical systems, and it leads to general path-space probability realizations of the initial transfer operator. The analytic data in our construction is a pair \((h,\lambda)\), where \(h\) is an \(R\)-harmonic function on \(X\), and \(\lambda\) is a given positive measure on \(X\) subject to a certain invariance condition defined from \(R\). With this we show that there are then discrete-time random-walk realizations in explicit path-space models; each associated to a probability measures \(\mathbb P\) on path-space, in such a way that the initial data allows for spectral characterization: The initial endomorphism in \(X\) lifts to an automorphism in path-space with the probability measure \(\mathbb P\) quasi-invariant with respect to a shift automorphism. The latter takes the form of explicit multiresolutions in \(L^2\) of \(\mathbb P\) in the sense of Lax-Phillips scattering theory.Optimal control for the nonlocal backward heat equationhttps://zbmath.org/1500.490132023-01-20T17:58:23.823708Z"Wang, Xiaoli"https://zbmath.org/authors/?q=ai:wang.xiaoli.2|wang.xiaoli.1|wang.xiaoli"He, Jinchun"https://zbmath.org/authors/?q=ai:he.jinchun"Xu, Haoyuan"https://zbmath.org/authors/?q=ai:xu.haoyuan"Yang, Meihua"https://zbmath.org/authors/?q=ai:yang.meihuaSummary: In this paper, we consider a nonlocal backward heat equation in a bounded domain \(\Omega\subset\mathbb{R}^n\) which has a Lipschitz boundary. The well-posedness is well known for the nonlocal forward heat equation, but the corresponding backward equation is ill-posed, at least in ordinary spaces. In this paper, we obtain the well-posedness of the nonlocal backward heat equation in a dense subspace of \(L^2(\Omega)\). Based on the well-posedness result, we obtain a singular optimal system (SOS) for the nonlocal backward heat system.
For the entire collection see [Zbl 1426.60001].Maximum number of pseudo diagonals with a minimum number of multiple crossings for convex polygonshttps://zbmath.org/1500.520042023-01-20T17:58:23.823708Z"Harborth, Heiko"https://zbmath.org/authors/?q=ai:harborth.heiko"Thürmann, Christian"https://zbmath.org/authors/?q=ai:thurmann.christianA convex \(n\)-gon has \(\binom{n}{4}\) crossings of diagonals. But what is the exact number of crossings if multiple crossings are only count once?
In general this number is unknown, and it seams easier to allow pseudo diagonals instead of straight lines. These pseudo diagonals are curved lines within the polygon connecting all pairs of vertices such that two lines with a common vertex have no crossing and any pair of two lines have at most one crossing.
The authors prove first, that the minimum number of crossings is \(n-3\) and that this minimum number can be reached in a unique way.
On the other side they prove that for \(n \geq 6\) the maximum number of
these extremal diagonals is given by \(D(n)\) where:
\[D(n) = \begin{cases}n+4 & \text{ if } n= 8,\\
n+3 & \text{ if } n= 6, 9,\\
n+2 & \text{ otherwise.}
\end{cases}
\]
Reviewer: Lienhard Wimmer (Isny)A stochastic approach to counting problemshttps://zbmath.org/1500.530462023-01-20T17:58:23.823708Z"Boulanger, Adrien"https://zbmath.org/authors/?q=ai:boulanger.adrienThe author studies orbital functions associated to finitely generated geometrically infinite Kleinian groups acting on the hyperbolic 3-space \(\mathbb{H}^3\).
Recall that the \textit{orbital function} associated to a group \(\Gamma\) which acts properly discontinuously on a metric space \((X, d)\) is defined by
\[
N_\Gamma(x,y,\rho) = \#\{ \gamma\in\Gamma, d(x,\gamma y) \leq \rho\},
\]
where \(x, y \in X\) and \(\rho > 0\). The growth of orbital functions of discrete groups was extensively studied since the 1950s.
The main results of the paper are the following two theorems.
\textbf{Theorem 1.2.} Let \(\Gamma\) be a degenerate Kleinian group such that the manifold \(M_\Gamma\) has positive injectivity radius and has both degenerate and geometrically finite ends. Then, for any \(x, y \in \mathbb{H}^3\) there are constants \(C > 0\) and \(\rho_0 > 0\) such that for any \(\rho > \rho_0\) we have
\[
N_\Gamma(x,y,\rho) \leq C\frac{e^{2\rho}}{\rho}.
\]
\textbf{Theorem 1.3.} Let \(\Gamma\) be a degenerate Kleinian group such that the manifold \(M_\Gamma\) has positive injectivity radius and the averaged orbital function is roughly decreasing, then
\(\bullet\) either all the ends of \(M_\Gamma\) are degenerate and there are constants \(C_-, C_+\) such that for any \(x, y \in \mathbb{H}^3\) there is \(\rho_0 > 0\) such that for \(\rho > \rho_0\) we have
\[
\frac{C_- e^{2\rho}}{\sqrt{\rho}} \leq N_\Gamma(x,y,\rho) \leq \frac{C_+ e^{2\rho}}{\sqrt{\rho}};
\]
or,
\(\bullet\) for any \(x, y \in M_\Gamma\) there are constants \(C_-, C_+, \rho_0 > 0\) such that for \(\rho > \rho_0\) we have
\[
\frac{C_- e^{2\rho}}{\rho^{\frac{3}{2}}} \leq N_\Gamma(x,y,\rho) \leq \frac{C_+ e^{2\rho}}{\rho^{\frac{3}{2}}}.
\]
Here a Kleinian group \(\Gamma\) is called \textit{degenerate} if it is finitely generated and the quotient manifold \(M_\Gamma = \mathbb{H}^3/\Gamma\) does not carry any finite Bowen-Margulis-Sullivan measure. The interest in degenerate groups comes from the fact that little was known about their orbital functions.
For the second theorem we recall that the \textit{averaged orbital function} is defined by \(\tilde{N}_{x,y}(\rho) = N(x,y,\rho)/e^{2\rho}\), and a function \(f: \mathbb{R}_+ \to \mathbb{R}_+\) is called \textit{roughly decreasing} if there exist positive constants \(C\) and \(\rho_0\) such that for any \(\rho_0 \leq \rho_1 \leq \rho_2\) one has \(f(\rho_2) \leq Cf(\rho_1)\).
The proofs of the theorems are based on the relation of the orbital function to the heat kernel, which is established using the drift property of the Brownian motion and the fact that \(\mathbb{H}^3\) is locally symmetric. On the way, the author gives some estimates for the orbital functions associated to nilpotent covers of compact hyperbolic manifolds, partially answering a question of M. Pollicott.
Reviewer: Mikhail Belolipetsky (Rio de Janeiro)On the functor of probability measures and quantization dimensionshttps://zbmath.org/1500.540032023-01-20T17:58:23.823708Z"Ivanov, A. V."https://zbmath.org/authors/?q=ai:ivanov.aleksandr-vladimirovich.1|ivanov.aleksandr-vladimirovich|ivanov.aleksandr-valentinovichSummary: The quantization dimensions of the probability measure given on the metric compact coincide with the dimensions of the finite approximation for the probability measure functor. Some functorial properties of quantization dimensions are established. It is shown that for any \(b>0\) there exists a metric compact \(X_b\) of capacitive dimension \(\text{dim}_{\text{B}}X_b = b\) on which there are probability measures with support equal to \(X\) whose quantization dimension takes all possible values from the interval \([0, b]\).On random weighted sum of positive semi-definite matriceshttps://zbmath.org/1500.600012023-01-20T17:58:23.823708Z"Galstyan, T. V."https://zbmath.org/authors/?q=ai:galstyan.t-v"Minasyan, A. G."https://zbmath.org/authors/?q=ai:minasyan.a-gSummary: Let \(A_1, \dots, A_n\) be fixed positive semi-definite matrices, i.e. \(A_i \in \mathbb{S}_p^+(\mathbb{R}) \forall i \in \{1, \dots, n\}\) and \(u_1, \dots, u_n\) are i.i.d. with \(u_i \sim \mathcal{N}(1, 1)\). Then, the object of our interest is the following probability \[\mathbb{P}\bigg(\sum_{i=1}^n u_i A_i \in \mathbb{S}_p^+(\mathbb{R})\bigg).\]
In this paper we examine this quantity for pairwise commutative matrices. Under some generic assumption about the matrices we prove that the weighted sum is also positive semi-definite with an overwhelming probability. This probability tends to 1 exponentially fast by the growth of number of matrices \(n\) and is a linear function with respect to the matrix dimension \(p\).Upper tails via high moments and entropic stabilityhttps://zbmath.org/1500.600022023-01-20T17:58:23.823708Z"Harel, Matan"https://zbmath.org/authors/?q=ai:harel.matan"Mousset, Frank"https://zbmath.org/authors/?q=ai:mousset.frank"Samotij, Wojciech"https://zbmath.org/authors/?q=ai:samotij.wojciechThis paper studies upper tail probability related to a bounded-degree polynomial with nonnegative coefficients on the \(p\)-based discrete hypercube. Let \(\Delta\ge 2\) be an integer and \(H\) be a connected non-bipartite and \(\Delta\)-regular graph. Suppose \(X\) is the number of copies of \(H\) in the random graph model \(G_{n,p}\). Let \(v_H\) be the number of vertices in \(H\). For every fixed \(\delta>0\) and \(p=p(n)\) satisfying \(n^{-1}(\ln n)^{\Delta v^2_H}\ll p^{\Delta/2}\ll 1\), it is shown that \(\lim_{n\rightarrow\infty}-\ln P(X\ge(1+\delta)EX)/(n^2p^{\Delta}\ln(1/p))=\delta^{2/v_H}/2\) if \(np^{\Delta}\rightarrow0\) and \(\min\{\delta^{2/v_H}/2,\theta\}\) if \(np^{\Delta}\rightarrow\infty\), where \(\theta\) is the unique positive solution to \(P_H(\theta)=1+\delta\). Here, \(P_H(\theta)=\sum_k i_k(H)\theta^k\), where \(i_k(H)\) is the number of independent sets of \(H\) of size \(k\). If \(H\) is a connected \(\Delta\)-regular graph, then for \(n^{-1}\ll p^{\Delta/2}\ll n^{-1}(\ln n)^{1/v_H^{-2}}\), it is shown that \(\lim_{n\rightarrow\infty}-\ln P(X\ge(1+\delta)EX)/EX=(1+\delta)\ln (1+\delta)-\delta\). Based on the developed techniques, cliques in random graphs and extensions to regular graphs have also be considered.
Reviewer: Yilun Shang (Newcastle upon Tyne)A central limit theorem for descents of a Mallows permutation and its inversehttps://zbmath.org/1500.600032023-01-20T17:58:23.823708Z"He, Jimmy"https://zbmath.org/authors/?q=ai:he.jimmyThis paper provides a limit theorem for some statistics of a random permutation on \(n\) elements which is not uniformly distributed but follows the Mallows distribution. This distribution is determined by a real parameter \(q> 0\) and the probability of permutation \(w \in S_n\) (where \(S_n\) denotes the symmetric group on \(n\) elements) is proportional to \(q^{\ell (w)}\), where \(\ell (w)\) denotes the number of inversions of \(w\). (Thus, \(\ell (w)\) is the number of pairs \(i<j\) such that \(w(i) > w(j)\).) The first result of the paper provides a central limit theorem for the random variable \(\mathrm{des} (w) + \mathrm{des} (w^{-1})\), where \(\mathrm{des} (w)\) is the number of descents of \(w\). This is the number of positions \(i \in [n-1]\) such that \(w(i+1) < w(i)\). In particular, the first theorem provides an upper bound on the distance between the expected value of any piecewise smooth function of the normalised \(\mathrm{des}(w) + \mathrm{des}(w^{-1})\) and the expected value of the same function applied to a standard normally distributed random variable. The second main theorem of the paper is about the asymptotic distribution of the rescaled vector \((\mathrm{des}(w), \mathrm{des}(w^{-1}))\). It shows that it converges to a normal bivariate distribution.
Reviewer: Nikolaos Fountoulakis (Birmingham)Random stable-type minimal factorizations of the \(n\)-cyclehttps://zbmath.org/1500.600042023-01-20T17:58:23.823708Z"Thévenin, Paul"https://zbmath.org/authors/?q=ai:thevenin.paulThis paper studies geometric representations of random factorizations of the permutation \((12\dots n)\) into a product of cycles \(\tau_1,\dots, \tau_k\) whose lengths \(\ell(\tau_i)\) satisfy the minimal condition \(\sum \ell(\tau_i)-1= n-1\). The case when all cycles are transpositions has been investigated by \textit{V. Féray} and \textit{I. Kortchemski} [Ann. Henri Lebesgue 1, 149--226 (2018; Zbl 1419.60008)] and a connection with the Brownian triangulation has been established. This paper extends this to the stable case when the factorization is sampled according to Boltzmann weights \(\prod w(\ell(\tau_i))\), which are fined-tuned so that polynomially large cycles appear when \(n\) is large. The key tools are various bijections with plane trees and their coding walks.
Reviewer: Nicolas Curien (Orsay)Growth of stationary Hastings-Levitovhttps://zbmath.org/1500.600052023-01-20T17:58:23.823708Z"Berger, Noam"https://zbmath.org/authors/?q=ai:berger.noam"Procaccia, Eviatar B."https://zbmath.org/authors/?q=ai:procaccia.eviatar-b"Turner, Amanda"https://zbmath.org/authors/?q=ai:turner.amanda-gSummary: We construct and study a stationary version of the Hastings-Levitov\((0)\) model. We prove that, unlike in the classical HL\((0)\) model, in the stationary case the size of particles attaching to the aggregate is tight, and therefore SHL\((0)\) is proposed as a potential candidate for a stationary off-lattice variant of diffusion limited aggregation (DLA). The stationary setting, together with a geometric interpretation of the harmonic measure, yields new geometric results such as stabilization, finiteness of arms and arm size distribution. We show that, under appropriate scaling, arms in SHL\((0)\) converge to the graph of Brownian motion which has fractal dimension \(3/2\). Moreover we show that trees with \(n\) particles reach a height of order \({n^{2/3}}\), corresponding to a numerical prediction of Meakin from 1983 for the gyration radius of DLA growing on a long line segment.Minimal matchings of point processeshttps://zbmath.org/1500.600062023-01-20T17:58:23.823708Z"Holroyd, Alexander E."https://zbmath.org/authors/?q=ai:holroyd.alexander-e"Janson, Svante"https://zbmath.org/authors/?q=ai:janson.svante"Wästlund, Johan"https://zbmath.org/authors/?q=ai:wastlund.johanSummary: Suppose that red and blue points form independent homogeneous Poisson processes of equal intensity in \({{\mathbb{R}}}^d\). For a positive (respectively, negative) parameter \(\gamma\) we consider red-blue matchings that locally minimize (respectively, maximize) the sum of \(\gamma\) th powers of the edge lengths, subject to locally minimizing the number of unmatched points. The parameter can be viewed as a measure of fairness. The limit \(\gamma \rightarrow -\infty\) is equivalent to Gale-Shapley stable matching. We also consider limits as \(\gamma\) approaches \(0, 1-, 1+\) and \(\infty \). We focus on dimension \(d=1\). We prove that almost surely no such matching has unmatched points. (This question is open for higher \(d)\). For each \(\gamma <1\) we establish that there is almost surely a unique such matching, and that it can be expressed as a finitary factor of the points. Moreover, its typical edge length has finite \(r\) th moment if and only if \(r<1/2\). In contrast, for \(\gamma =1\) there are uncountably many matchings, while for \(\gamma >1\) there are countably many, but it is impossible to choose one in a translation-invariant way. We obtain existence results in higher dimensions (covering many but not all cases). We address analogous questions for one-colour matchings also.Probability distributions characterisations on a homogeneous conehttps://zbmath.org/1500.600072023-01-20T17:58:23.823708Z"Boutouria, Imen"https://zbmath.org/authors/?q=ai:boutouria.imen"Bouzida, Imed"https://zbmath.org/authors/?q=ai:bouzida.imedIn this paper, the authors study probability distributions on a homogeneous cone and generalize Gindikin results to Vinberg algebras.
Reviewer: Alessandro Selvitella (Fort Wayne)On the density for sums of independent exponential, Erlang and gamma variateshttps://zbmath.org/1500.600082023-01-20T17:58:23.823708Z"Levy, Edmond"https://zbmath.org/authors/?q=ai:levy.edmondThe paper begins with the formula for the hypo-exponential density for the sum of independent exponentials having pairwise distinct parameters.
The author points out that this density has a divided difference characteristic which immediately suggests a novel perspective to further explore the densities of sums of independent exponentials.
The paper advances a succinct representation for the density of independent Erlang distributed variables and demonstrates agreement with others papers where such formulae have been found (and often rediscovered) by other means.
The author extends these results further, by using the tools of fractional calculus (see for example [\textit{K. B. Oldham} and \textit{J. Spanier}, The fractional calculus. Theory and applications of differentiation and integration to arbitrary order. Elsevier, Amsterdam (1974; Zbl 0292.26011)]), a representation is also founded for the density for sums of distinct independent gamma random variables. The paper concludes by showing how this approach produces the density function itself.
Reviewer: Romeo Negrea (Timişoara)A probabilistic interpretation of the Dzhrbashyan fractional integralhttps://zbmath.org/1500.600092023-01-20T17:58:23.823708Z"Zhao, Dazhi"https://zbmath.org/authors/?q=ai:zhao.dazhi"Yu, Guozhu"https://zbmath.org/authors/?q=ai:yu.guozhu"Yu, Tao"https://zbmath.org/authors/?q=ai:yu.tao"Zhang, Lu"https://zbmath.org/authors/?q=ai:zhang.luSummary: Physical and probabilistic interpretations of the fractional derivatives and integrals are basic problems to their applications. In this paper, we establish a relation between the Dzhrbashyan fractional integral and the expectation of a corresponding random variable by constructing the cumulative distribution function. As examples, interpretations of the Riemann-Liouville fractional integral and Kober integral operator are given. Furthermore, probabilistic interpretations of the Caputo fractional derivative and the fractional integral of a function with respect to another function are discussed too. With the help of probabilistic interpretations proposed in this paper, models described by fractional derivatives and integrals can be endowed with corresponding statistical meanings, while some statistical physics models can be rewritten in fractional calculus too.Lorenz and polarization orderings of the double-Pareto lognormal distribution and other size distributionshttps://zbmath.org/1500.600102023-01-20T17:58:23.823708Z"Okamoto, Masato"https://zbmath.org/authors/?q=ai:okamoto.masatoSummary: Polarization indices such as the Foster-Wolfson index have been developed to measure the extent of clustering in a few classes with wide gaps between them in terms of income distribution. However, \textit{X. Zhang} and \textit{Kanbur} [``What difference do polarization measures make? An application to China'', J. Dev. Stud. 37, 85--98 (2001)] failed to empirically find clear differences between polarization and inequality indices in the measurement of intertemporal distributional changes. This paper addresses this `distinction' problem on the level of the respective underlying stochastic orders, the polarization order (PO) in distributions divided into two nonoverlapping classes and the Lorenz order (LO) of inequality in distributions. More specifically, this paper investigates whether a distribution \(F\) can be either more or less polarized than a distribution \(H\) in terms of the PO if \(F\) is more unequal than \(H\) in terms of the LO. Furthermore, this paper derives conditions for the LO and PO of the double-Pareto lognormal (dPLN) distribution. The derived conditions are applicable to sensitivity analyses of inequality and polarization indices with respect to distributional changes. From this application, a suggestion for appropriate two-class polarization indices is made.On Brascamp-Lieb and Poincaré type inequalities for generalized tempered stable distributionhttps://zbmath.org/1500.600112023-01-20T17:58:23.823708Z"Barman, Kalyan"https://zbmath.org/authors/?q=ai:barman.kalyan"Upadhye, Neelesh S."https://zbmath.org/authors/?q=ai:upadhye.neelesh-sIn this paper, the authors analyze covariance inequalities using Stein's method for the class of generalized tempered stable probability distributions. They obtain a Stein's lemma and derive a Stein operator for this class. They also prove Brascamp-Lieb and Poincaré type inequalities for the class of generalized tempered stable probability distributions.
Reviewer: Alessandro Selvitella (Fort Wayne)On the central limit theorem for stationary random fields under \({\mathbb{L}^1}\)-projective conditionhttps://zbmath.org/1500.600122023-01-20T17:58:23.823708Z"Lin, Han-Mai"https://zbmath.org/authors/?q=ai:lin.han-mai"Merlevède, Florence"https://zbmath.org/authors/?q=ai:merlevede.florence"Volný, Dalibor"https://zbmath.org/authors/?q=ai:volny.daliborSummary: The first aim of this paper is to wonder to what extent we can generalize the central limit theorem of \textit{M. I. Gordin} [``Abstracts of communication'', in: T.1 A-K, International Conference on Probability Theory, Vilnius (1973)] under the so-called \({\mathbb{L}^1}\)-projective criteria to ergodic stationary random fields when completely commuting filtrations are considered. Surprisingly it appears that this result cannot be extended to its full generality and that an additional condition is needed.Central limit theorem for bifurcating Markov chains under pointwise ergodic conditionshttps://zbmath.org/1500.600132023-01-20T17:58:23.823708Z"Penda, S. Valère Bitseki"https://zbmath.org/authors/?q=ai:bitseki-penda.s-valere"Delmas, Jean-François"https://zbmath.org/authors/?q=ai:delmas.jean-francoisSummary: Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for general additive functionals of BMC, and prove the existence of three regimes. This corresponds to a competition between the reproducing rate (each individual has two children) and the ergodicity rate for the evolution of the trait. This is in contrast with the work of \textit{J. Guyon} [Ann. Appl. Probab. 17, No. 5--6, 1538--1569 (2007; Zbl 1143.62049)], where the considered additive functionals are sums of martingale increments, and only one regime appears. Our result can be seen as a discrete time version, but with general trait evolution, of results in the time continuous setting of branching particle system from [\textit{R. Adamczak} and \textit{P. Miłoś}, Electron. J. Probab. 20, Paper No. 42, 35 p. (2015; Zbl 1321.60035)], where the evolution of the trait is given by an Ornstein-Uhlenbeck process.A CLT for second difference estimators with an application to volatility and intensityhttps://zbmath.org/1500.600142023-01-20T17:58:23.823708Z"Stoltenberg, Emil A."https://zbmath.org/authors/?q=ai:stoltenberg.emil-aas"Mykland, Per A."https://zbmath.org/authors/?q=ai:mykland.per-aslak"Zhang, Lan"https://zbmath.org/authors/?q=ai:zhang.lanSummary: In this paper, we introduce a general method for estimating the quadratic covariation of one or more spot parameter processes associated with continuous time semimartingales, and present a central limit theorem that has this class of estimators as one of its applications. The class of estimators we introduce, that we call Two-Scales Quadratic Covariation \((\text{TSQC})\) estimators, is based on sums of increments of second differences of the observed processes, and the intervals over which the differences are computed are rolling and overlapping. This latter feature lets us take full advantage of the data, and, by sufficiency considerations, ought to outperform estimators that are based on only one partition of the observational window. Moreover, a two-scales approach is employed to deal with asymptotic bias terms in a systematic manner, thus automatically giving consistent estimators without having to work out the form of the bias term on a case-to-case basis. We highlight the versatility of our central limit theorem by applying it to a novel leverage effect estimator that does not belong to the class of \(\text{TSQC}\) estimators. The principal empirical motivation for the present study is that the discrete times at which a continuous time semimartingale is observed might depend on features of the observable process other than its level, such as its spot-volatility process. As an application of the \(\text{TSQC}\) estimators, we therefore show how it may be used to estimate the quadratic covariation between the spot-volatility process and the intensity process of the observation times, when both of these are taken to be semimartingales. The finite sample properties of this estimator are studied by way of a simulation experiment, and we also apply this estimator in an empirical analysis of the Apple stock. Our analysis of the Apple stock indicates a rather strong correlation between the spot volatility process of the log-prices process and the times at which this stock is traded and hence observed.Large deviations for spectral measures of some spiked matriceshttps://zbmath.org/1500.600152023-01-20T17:58:23.823708Z"Noiry, Nathan"https://zbmath.org/authors/?q=ai:noiry.nathan"Rouault, Alain"https://zbmath.org/authors/?q=ai:rouault.alainIn this paper, the authors consider three classical models of random matrices:
the Hermite ensemble (i.e., a probability distribution on Hermitian matrices~\(X_n\) of size \({n\times n}\), whose density is proportional to \({\exp(-1/2\mathrm{Tr}(X_nX^*_n))}\) with respect to the Lebesgue measure \(dX_n\)); the Laguerre ensemble (i.e., a distribution of \({X_nX^*_n}\) with density proportional to \(\det(X_nX_n^*)^{N-n}\exp(-\mathrm{Tr} X_nX^*_n),\) where \(X_n\) is a \({n\times N}\) complex matrix with entries whose real and imaginary parts are i.i.d. \({\mathcal{N}(0; 1/2)}\), and \({N>n}\));
the Gross-Witten ensemble with parameter \({g\in\mathbb{R}}\) (i.e., a probability measure on the unitary group \(U(n)\) whose density is proportional to \({\exp(\frac{ng}{2}\mathrm{Tr}(U+U^*))}\) with respect to the Haar measure \(dU\)).
The authors establish large deviations principles for spectral measures of rank-one perturbations of these models. The proofs rely on the large deviations results for unperturbed models (see [\textit{F. Gamboa} et al., J. Funct. Anal. 270, No. 2, 509--559 (2016; Zbl 1327.47028)]). One of two approaches uses also the computations of the Jacobi and Verblunsky coefficients of the deformed models.
Reviewer: Ivan Podvigin (Novosibirsk)Large deviations for stochastic \(2D\) Navier-Stokes equations on time-dependent domainshttps://zbmath.org/1500.600162023-01-20T17:58:23.823708Z"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.2|wang.wei.9|wang.wei.166|wang.wei.20|wang.wei.24|wang.wei.30|wang.wei.3|wang.wei.12|wang.wei.8|wang.wei.1|wang.wei.13|wang.wei.21"Zhai, Jianliang"https://zbmath.org/authors/?q=ai:zhai.jianliang"Zhang, Tusheng"https://zbmath.org/authors/?q=ai:zhang.tusheng-sSummary: A Freidlin-Wentzell-type large deviation principle is established for \(2D\) stochastic Navier-Stokes equations on time-dependent domains driven by Brownian motion, which captures situations where the regions of the fluid change with time.Local and uniform moduli of continuity of chi-square processeshttps://zbmath.org/1500.600172023-01-20T17:58:23.823708Z"Marcus, Michael B."https://zbmath.org/authors/?q=ai:marcus.michael-b"Rosen, Jay"https://zbmath.org/authors/?q=ai:rosen.jay-sThe authors show that when the Gaussian process \(\eta\) has a local or uniform modulus of continuity the related \(k/2\) chi-square process has a closely related local or uniform modulus of continuity.
The main results are given in Theorem 1.1 (which seems to extend a similar result given by the authors [Asymptotic properties of permanental sequences. Related to birth and death processes and autoregressive Gaussian sequences. Cham: Springer (2021; Zbl 1486.60001)]) and in Theorem 1.2 (this result extend a good result given by the same authors in [Markov processes, Gaussian processes, and local times. Cambridge: Cambridge University Press (2006; Zbl 1129.60002)]).
Reviewer: Romeo Negrea (Timişoara)The multivariate functional de Jong CLThttps://zbmath.org/1500.600182023-01-20T17:58:23.823708Z"Döbler, Christian"https://zbmath.org/authors/?q=ai:dobler.christian"Kasprzak, Mikołaj"https://zbmath.org/authors/?q=ai:kasprzak.mikolaj-j"Peccati, Giovanni"https://zbmath.org/authors/?q=ai:peccati.giovanniSummary: We prove a multivariate functional version of \textit{P. de Jong}'s CLT [J. Multivariate Anal. 34, No. 2, 275--289 (1990; Zbl 0709.60019)] yielding that, given a sequence of vectors of Hoeffding-degenerate U-statistics, the corresponding empirical processes on \([0, 1]\) weakly converge in the Skorohod space as soon as their fourth cumulants in \(t=1\) vanish asymptotically and a certain strengthening of the Lindeberg-type condition is verified. As an application, we lift to the functional level the `universality of Wiener chaos' phenomenon first observed in [\textit{I. Nourdin} et al., Ann. Probab. 38, No. 5, 1947--1985 (2010; Zbl 1246.60039)].Introducing smooth amnesia to the memory of the elephant random walkhttps://zbmath.org/1500.600192023-01-20T17:58:23.823708Z"Laulin, Lucile"https://zbmath.org/authors/?q=ai:laulin.lucileSummary: This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the AERW.The Fokker-Planck equation for the time-changed fractional Ornstein-Uhlenbeck stochastic processhttps://zbmath.org/1500.600202023-01-20T17:58:23.823708Z"Ascione, Giacomo"https://zbmath.org/authors/?q=ai:ascione.giacomo"Mishura, Yuliya"https://zbmath.org/authors/?q=ai:mishura.yuliya-s"Pirozzi, Enrica"https://zbmath.org/authors/?q=ai:pirozzi.enricaThe authors study properties of the generalized Fokker-Planck equation induced by the time-changed fractional Ornstein-Uhlenbeck process. They introduce inverse subordinators, some concepts from generalized fractional calculus and the subordination and weighted subordination operators. They give a sufficient condition for differentiability of subordinated functions linked to the inverse \(\alpha\)-stable subordinator. The problems of uniqueness of mild solution and of strong solution are studied.
Reviewer: B. L. S. Prakasa Rao (Hyderabad)Random walks on hyperbolic spaces: concentration inequalities and probabilistic Tits alternativehttps://zbmath.org/1500.600212023-01-20T17:58:23.823708Z"Aoun, Richard"https://zbmath.org/authors/?q=ai:aoun.richard"Sert, Cagri"https://zbmath.org/authors/?q=ai:sert.cagriSummary: The goal of this article is two-fold: in a first part, we prove Azuma-Hoeffding type concentration inequalities around the drift for the displacement of non-elementary random walks on hyperbolic spaces. For a proper hyperbolic space \(M\), we obtain explicit bounds that depend only on \(M\), the size of support of the measure as in the classical case of sums of independent random variables, and on the norm of the driving probability measure in the left regular representation of the group of isometries. We obtain uniform bounds in the case of hyperbolic groups and effective bounds for simple linear groups of rank-one. In a second part, using our concentration inequalities, we give quantitative finite-time estimates on the probability that two independent random walks on the isometry group of a hyperbolic space generate a free non-abelian subgroup. Our concentration results follow from a more general, but less explicit statement that we prove for cocycles which satisfy a certain cohomological equation. For example, this also allows us to obtain subgaussian concentration bounds around the top Lyapunov exponent of random matrix products in arbitrary dimension.Conservative random walkhttps://zbmath.org/1500.600222023-01-20T17:58:23.823708Z"Engländer, János"https://zbmath.org/authors/?q=ai:englander.janos"Volkov, Stanislav"https://zbmath.org/authors/?q=ai:volkov.stanislavSummary: Recently, in [\textit{J. Engländer} et al., Electron. J. Probab. 25, Paper No. 3, 38 p. (2020; Zbl 1467.60031)], the ``coin-turning walk'' was introduced on \(\mathbb{Z} \). It is a non-Markovian process where the steps form a (possibly) time-inhomogeneous Markov chain. In this article, we follow up the investigation by introducing analogous processes in \({\mathbb{Z}^d},\, d\ge 2\): at time \(n\) the direction of the process is ``updated'' with probability \({p_n} \); otherwise the next step repeats the previous one. We study some of the fundamental properties of these walks, such as transience/recurrence and scaling limits.
Our results complement previous ones in the literature about ``correlated'' (or ``Newtonian'') and ``persistent'' random walks.A new class of multivariate counting processes and its characterizationhttps://zbmath.org/1500.600232023-01-20T17:58:23.823708Z"Cha, Ji Hwan"https://zbmath.org/authors/?q=ai:cha.ji-hwan"Giorgio, Massimiliano"https://zbmath.org/authors/?q=ai:giorgio.massimilianoSummary: In this paper, we suggest a new class of multivariate counting processes which generalizes and extends the multivariate generalized Polya process recently studied in [the authors, Adv. Appl. Probab. 48, No. 2, 443--462 (2016; Zbl 1346.60066)]. Initially, we define this multivariate counting process by means of mixing. For further characterization of it, we suggest an alternative definition, which facilitates convenient characterization of the proposed process. We also discuss the dependence structure of the proposed multivariate counting process and other stochastic properties such as the joint distributions of the number of events in an arbitrary interval or disjoint intervals and the conditional joint distribution of the arrival times of different types of events given the number of events. The corresponding marginal processes are also characterized.Partial isometries, duality, and determinantal point processeshttps://zbmath.org/1500.600242023-01-20T17:58:23.823708Z"Katori, Makoto"https://zbmath.org/authors/?q=ai:katori.makoto"Shirai, Tomoyuki"https://zbmath.org/authors/?q=ai:shirai.tomoyukiLet \(\mathbf{S}\) be a locally compact Hausdorff space with countable base, and \(\lambda\) be a Radon measure on \(\mathbf{S}\). A determinantal point process is an ensemble of random nonnegative-integer-valued Radon measures \(\Xi\) on a space \(\mathbf{S}\), whose correlation functions are all given by determinants specified by an integral kernel \(K\) called the correlation kernel. Let \(\mathcal{K}\) be the integral projection operator on \(L^2(\mathbf{S}, \lambda)\) with kernel \(K\). In the present paper, the case in which \(\mathcal{K} f= f\) for all \(f\in(\ker\mathcal{K})^{\perp}\subset L^2(\mathbf{S}, \lambda)\) is considered, where \((\ker\mathcal{K})^{\perp}\) denotes the orthogonal complement of the kernel space of \(\mathcal{K}\). That is, \(\mathcal{K}\) is an orthogonal projection. By definition, it is obvious that the condition \(0\le\mathcal{K} \le I\) is satisfied. The purpose of the present paper is to propose a useful method to provide orthogonal projections \(\mathcal{K}\) and determinantal point processes whose correlation kernels are given by the Hermitian integral kernels of \(\mathcal{K}\), \(K(x, x')\), \(x, x'\in\mathbf{S}\). The authors consider a pair of Hilbert spaces, \(H_l\), \(l = 1, 2\), which are assumed to be realized as \(L^2\)-spaces, \(L^2(\mathbf{S}_l, \lambda_l)\), \(l = 1, 2\), and introduce a bounded linear operator \(\mathcal{W}\): \(H_1 \to H_2\) and its adjoint \(\mathcal{W}^*\): \(H_2 \to H_1\). The paper is organized as follows. In Section 2, the authors give the main theorems which enable to generate determinantal point processes. Sections 3 and 4 are devoted to a variety of examples of determinantal point processes obtained by the framework of the paper for the one-dimensional and the two-dimensional spaces, respectively. Examples in spaces with arbitrary dimensions \(d\in\mathbb{N}\) are given in Section 5. The authors list out open problems in Section 6. Appendices A and C are used to explain useful multivariate functions and determinantal formulas associated with the classical and the affine root systems, respectively. The definitions and basic properties of the Jacobi theta functions are summarized in Appendix B.
Reviewer: Viktor Ohanyan (Erevan)Rigidity of determinantal point processes on the unit disc with sub-Bergman kernelshttps://zbmath.org/1500.600252023-01-20T17:58:23.823708Z"Qiu, Yanqi"https://zbmath.org/authors/?q=ai:qiu.yanqi"Wang, Kai"https://zbmath.org/authors/?q=ai:wang.kai.4|wang.kai.1|wang.kai|wang.kai.3|wang.kai.2The definition of number rigidity is due to [\textit{S. Ghosh}, Probab. Theory Relat. Fields 163, No. 3--4, 643--665 (2015; Zbl 1334.60083)], where the author showed that the sine-process is number rigid and [\textit{S. Ghosh} and \textit{Y. Peres}, Duke Math. J. 166, No. 10, 1789--1858 (2017; Zbl 1405.60067)], where the authors showed that the Ginibre process and the zero set of Gaussian analytic functions on the plane are number rigid. \textit{A. I. Bufetov} [Bull. Math. Sci. 6, No. 1, 163--172 (2016; Zbl 1335.60075)] proved that determinantal point processes with the Airy, the Bessel and the Gamma kernels are number rigid. For more results on the number rigidity of point processes, see [\textit{S. Ghosh} and \textit{J. L. Lebowitz}, Indian J. Pure Appl. Math. 48, No. 4, 609--631 (2017; Zbl 1390.60104)]. However, \textit{A. E. Holroyd} and \textit{T. Soo} [Electron. J. Probab. 18, Paper No. 74, 24 p. (2013; Zbl 1291.60101)] showed that the determinantal point process on the unit disc \(\mathcal D\) with the standard Bergman kernel (with respect to the normalized Lebesgue measure on the unit disc \(\mathcal{D}\): \(K_{\mathcal{D}}(z,\omega) = \frac{1}{(1- z \bar{\omega})^2} = \sum_{n=0}^{\infty}(n + 1)\, z^n \bar{\omega}^n\) is not number rigid. More generally, in [\textit{A. I. Bufetov} et al., Probab. Theory Relat. Fields 172, No. 1--2, 31--69 (2018; Zbl 1429.60047)] it is shown that for any domain \(U\) in the \(d\)-dimensional complex Euclidean space \(\mathbb{C}^d\) without Liouville property and any weight \(\omega: U \to\mathbb{R}^+\) locally uniformly away from zero, the determinantal point process associated with the reproducing kernel of the weighted Bergman space \(L_a^2(U,\omega)\) is not number rigid. These results lead to ask whether there exist natural number rigid determinantal point processes on a bounded domain of the complex plane (of course, any finite rank orthogonal projection yields a number rigid determinantal point process, so the authors of the present paper are only interested in infinite rank orthogonal projections).
In the present paper, the authors answer affirmatively this question in both deterministic and probabilistic ways.
Reviewer: Viktor Ohanyan (Erevan)On stationarity properties of generalized Hermite-type processeshttps://zbmath.org/1500.600262023-01-20T17:58:23.823708Z"Donhauzer, Illia"https://zbmath.org/authors/?q=ai:donhauzer.illia"Olenko, Andriy"https://zbmath.org/authors/?q=ai:olenko.andriy-yaThe paper is devoted to the study of properties of limit processes in non-central limit theorems for nonlinear integral functionals. Namely, the authors investigate generalized Hermite-type processes obtained via asymptotics of nonlinear transformations of long-range dependent random fields. The main result is that contrary to the classical one-dimensional case of stochastic processes, integral functionals of nonlinear transformations of long-range dependent random fields converge to the generalized Hermite-type process with non-stationary increments. Moreover, relationships between the increments of the limit processes and geometric probabilities are shown.
Reviewer: Utkir A. Rozikov (Tashkent)Exponential tightness for integral-type functionals of centered independent differently distributed random variableshttps://zbmath.org/1500.600272023-01-20T17:58:23.823708Z"Logachev, Artem Vasilhevich"https://zbmath.org/authors/?q=ai:logachev.artem-vasilhevich"Mogulskii, Anatolii Alfredovich"https://zbmath.org/authors/?q=ai:mogulskii.a-aA sequence of random elements \(\mathcal{F}_{n} \in \mathbb{X}, n \in \mathbb{N}\) is called exponentially tight in a metric space \(\mathbb{X}\) with a normalizing function \(\psi(n): \lim _{n \rightarrow \infty} \psi(n)=\) \(\infty\), if for any \(C>0\) there exists a compact \(K_{C} \subseteq \mathbb{X}\) such that
\[
\limsup _{n \rightarrow \infty} \frac{1}{\psi(n)} \ln \mathbf{P}\left(\mathcal{F}_{n} \notin K_{C}\right) \leq-C.
\]
In this paper, the exponential tightness is proved for a sequence of integral type random fields constructed by centered independent differently distributed random variables. To do this the authors apply sufficient conditions for the exponential tightness of a sequence of continuous random fields.
Reviewer: Utkir A. Rozikov (Tashkent)The tail process and tail measure of continuous time regularly varying stochastic processeshttps://zbmath.org/1500.600282023-01-20T17:58:23.823708Z"Soulier, Philippe"https://zbmath.org/authors/?q=ai:soulier.philippeAuthor's abstract: The goal of this paper is to investigate the tools of extreme value theory originally introduced for discrete time stationary stochastic processes (time series), namely the tail process and the tail measure, in the framework of continuous time stochastic processes with paths in the space \(\mathcal{D}\) of càdlàg functions indexed by \(\mathbb{R}\), endowed with Skorohod's \(J_1\) topology. We prove that the essential properties of these objects are preserved, with some minor (though interesting) differences arising. We first obtain structural results which provide representation for homogeneous shift-invariant measures on \(\mathcal{D}\) and then study regular variation of random elements in \(\mathcal{D}\). We give practical conditions and study several examples, recovering and extending known results.
Reviewer: Fabrizio Durante (Lecce)Model-adaptive optimal discretization of stochastic integralshttps://zbmath.org/1500.600292023-01-20T17:58:23.823708Z"Gobet, Emmanuel"https://zbmath.org/authors/?q=ai:gobet.emmanuel"Stazhynski, Uladzislau"https://zbmath.org/authors/?q=ai:stazhynski.uladzislauSummary: We study the optimal discretization error of stochastic integrals driven by a multidimensional continuous Brownian semimartingale. In the previous works a pathwise lower bound for the renormalized quadratic variation of the error was provided together with an asymptotically optimal discretization strategy, i.e. for which the lower bound is attained. However the construction of the optimal strategy involved the knowledge about the diffusion coefficient of the semimartingaleunder study. In this work we provide a model-adaptive asymptotically optimal discretization strategy that does not require any prior knowledge about the model. We prove the optimality for quite general class of discretization strategies based on kernel techniques for adaptive estimation and previously obtained optimal strategies that use random ellipsoid hitting times.Approximation to stochastic variance reduced gradient Langevin dynamics by stochastic delay differential equationshttps://zbmath.org/1500.600302023-01-20T17:58:23.823708Z"Chen, Peng"https://zbmath.org/authors/?q=ai:chen.peng"Lu, Jianya"https://zbmath.org/authors/?q=ai:lu.jianya"Xu, Lihu"https://zbmath.org/authors/?q=ai:xu.lihuThe authors consider the stochastic variance reduced gradient Langevin dynamics algorithm, and show that its output can be approximated by the solution of a stochastic differential equations with delay. Uniform error bounds are obtained using the Wasserstein distance under assumptions that cover both convex and non-convex minimization problems. The proof arguments rely on the Lindeberg principle and on semigroup gradient bounds obtained by the Malliavin calculus.
Reviewer: Nicolas Privault (Singapore)Asymptotic expansion for forward-backward SDEs with jumpshttps://zbmath.org/1500.600312023-01-20T17:58:23.823708Z"Fujii, Masaaki"https://zbmath.org/authors/?q=ai:fujii.masaaki"Takahashi, Akihiko"https://zbmath.org/authors/?q=ai:takahashi.akihikoSummary: This work provides a semi-analytic approximation method for decoupled forward-backward SDEs (FBSDEs) with jumps. In particular, we construct an asymptotic expansion method for FBSDEs driven by the random Poisson measures with \(\sigma \)-finite compensators as well as the standard Brownian motions around the small-variance limit of the forward SDE. We provide a semi-analytic solution technique as well as its error estimate for which we only need to solve essentially a system of linear ODEs. In the case of a finite jump measure with a bounded intensity, the method can also handle state-dependent and hence non-Poissonian jumps, which are quite relevant for many practical applications.Strong solutions of stochastic differential equations with coefficients in mixed-norm spaceshttps://zbmath.org/1500.600322023-01-20T17:58:23.823708Z"Ling, Chengcheng"https://zbmath.org/authors/?q=ai:ling.chengcheng"Xie, Longjie"https://zbmath.org/authors/?q=ai:xie.longjieBy studying the parabolic equations in mixed-norm spaces, the authors prove the existence and uniqueness of strong solutions to stochastic differential equations driven by Brownian motion with coefficients in spaces with mixed-norm. Such mixed-norm spaces are necessary when the physical processes have different behavior with respect to each component, and the advantage lies in the flexible integrability of the coefficients.
Reviewer: Longjie Xie (Xuzhou)Optimal sampling design for global approximation of jump diffusion stochastic differential equationshttps://zbmath.org/1500.600332023-01-20T17:58:23.823708Z"Przybyłowicz, Paweł"https://zbmath.org/authors/?q=ai:przybylowicz.pawelSummary: The paper deals with strong global approximation of stochastic differential equations (SDEs) driven by two independent processes: a nonhomogeneous Poisson process and a Wiener process. We assume that the jump and diffusion coefficients of the underlying SDE satisfy jump commutativity condition (see Chapter 6.3 in [\textit{E. Platen} and \textit{N. Bruti-Liberati}, Numerical solution of stochastic differential equations with jumps in finance. Berlin: Springer (2010; Zbl 1225.60004)]). We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson and Wiener processes. We consider classes of methods that use equidistant or nonequidistant sampling of the Poisson and Wiener processes. We provide a construction of optimal methods, based on the classical Milstein scheme, which asymptotically attain the established minimal errors. The analysis implies that methods based on nonequidistant mesh are more efficient, with respect to asymptotic constants, than those based on the equidistant mesh.On exponential stability of non-autonomous stochastic differential equations with Markovian switchinghttps://zbmath.org/1500.600342023-01-20T17:58:23.823708Z"Tran, Ky Q."https://zbmath.org/authors/?q=ai:tran.ky-quan"Le, Bich T. N."https://zbmath.org/authors/?q=ai:le.bich-t-nSummary: This paper is devoted to exponential stability of a class of non-autonomous stochastic differential equations with Markovian switching. By making use the time inhomogeneous property of the drift and diffusion coefficients, we derive sufficient and verifiable conditions for moment exponential stability and almost sure exponential stability. The contribution of the Markovian switching and time-inhomogeneous property to the stability is revealed. Two examples are provided to illustrate the effectiveness of our criteria.On approximations of the Euler-Peano scheme for multivalued stochastic differential equationshttps://zbmath.org/1500.600352023-01-20T17:58:23.823708Z"Wu, Jing"https://zbmath.org/authors/?q=ai:wu.jing|wu.jing.1"Zhang, Mingbo"https://zbmath.org/authors/?q=ai:zhang.mingboSummary: It is known that a unique strong solution exists for multivalued stochastic differential equations under the Lipschitz continuity and linear growth conditions. In this paper we apply the Euler-Peano scheme to show that existence of weak solution and pathwise uniqueness still hold when the coefficients are random and satisfy one-sided locally Lipschitz continuous and an integral condition (i.e. [\textit{N. V. Krylov} et al., Stochastic PDE's and Kolmogorov equations in infinite dimensions. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (CIME) held in Cetraro, Italy, 1998. Berlin: Springer (1999; Zbl 0927.00037)]). When the coefficients are nonrandom and possibly discontinuous but only satisfy some integral conditions, the sequence of solutions of the Euler-Peano scheme converges weakly, and the limit is a weak solution of the corresponding MSDE. As a particular case, we obtain a global semi-flow for stochastic differential equations reflected in closed, convex domains.Unique strong solutions of Lévy processes driven stochastic differential equations with discontinuous coefficientshttps://zbmath.org/1500.600362023-01-20T17:58:23.823708Z"Xiong, Jie"https://zbmath.org/authors/?q=ai:xiong.jie"Zheng, Jiayu"https://zbmath.org/authors/?q=ai:zheng.jiayu"Zhou, Xiaowen"https://zbmath.org/authors/?q=ai:zhou.xiaowenSummary: We study the strong solutions for a class of one-dimensional stochastic differential equations driven by a Brownian motion and a pure jump Lévy process. Under fairly general conditions on the coefficients, we prove the pathwise uniqueness by showing the weak uniqueness and applying a local time technique.Spatial average for the solution to the heat equation with Rosenblatt noisehttps://zbmath.org/1500.600372023-01-20T17:58:23.823708Z"Dhoyer, Rémi"https://zbmath.org/authors/?q=ai:dhoyer.remi"Tudor, Ciprian A."https://zbmath.org/authors/?q=ai:tudor.ciprian-aLet \(Z_{(H_0, \mathbf{H}),d+1}(t,x)\) be the Rosenblatt sheet with Hurst parameter \((H_0,\mathbf{H})\in (\frac 1 2,1)^{d+1},\) and consider the stochastic heat equation on \(\mathbb R^d\):
\[
\partial_t u(t,x) = \frac 1 2 \Delta u(t,x) + \dot Z_{(H_0,\mathbf{H}),d+1}(t,x),\ \ t\ge 0, x\in\mathbb R^d.
\]
By analyzing the cumulants of the solution, it is proved that as \(R\to\infty\) the spatial average
\[
G_R(t):= R^{-(H_1+\cdots+H_d)}\int_{[-R,R]^d} \big\{u(t,x)-1\big\} d x
\]
converges weakly, in the space of continuous functions, to the Rosenblatt process \(Z_{(H_0,\mathbf{H}),d+1}(t,\mathbf{2})\).
Reviewer: Feng-Yu Wang (Tianjin)Nontrivial equilibrium solutions and general stability for stochastic evolution equations with pantograph delay and tempered fractional noisehttps://zbmath.org/1500.600382023-01-20T17:58:23.823708Z"Liu, Yarong"https://zbmath.org/authors/?q=ai:liu.yarong"Wang, Yejuan"https://zbmath.org/authors/?q=ai:wang.yejuan"Caraballo, Tomas"https://zbmath.org/authors/?q=ai:caraballo.tomasSummary: In this paper, we investigate the asymptotic behavior of stochastic pantograph delay evolution equations driven by a tempered fractional Brownian motion (tfBm) with Hurst parameter \(H > 1 /2\). First of all, the global existence, uniqueness, and mean-square stability with general decay rate of mild solutions are established. In particular, we would like to point out that our analysis is not necessary to construct Lyapunov functions, but we deal directly with stability via the Banach fixed point theorem, the fractional power of operators, and the semigroup theory. It is worth emphasizing that a novel estimate of stochastic integrals with respect to tfBm is presented, which greatly contributes to the stability analyses. Then after extending the factorization formula to the tfBm case, we construct the nontrivial equilibrium solution, defined for \(t\in \mathbb{R}\), by means of an approximation technique and a convergence analysis. Moreover, we analyze the Hölder regularity in time and general stability (including both polynomial and logarithmic stability) of the nontrivial equilibrium solution in the sense of mean-square. As an example of application, the reaction diffusion neural network system with pantograph delay is considered, and the nontrivial equilibrium solution and general stability of the system are proved under the Lipschitz assumption.Singular HJB equations with applications to KPZ on the real linehttps://zbmath.org/1500.600392023-01-20T17:58:23.823708Z"Zhang, Xicheng"https://zbmath.org/authors/?q=ai:zhang.xicheng|zhang.xicheng.1"Zhu, Rongchan"https://zbmath.org/authors/?q=ai:zhu.rongchan"Zhu, Xiangchan"https://zbmath.org/authors/?q=ai:zhu.xiang-chanThis paper focuses on the Hamilton-Jacobi-Bellman equations with distribution valued coefficients, which are not well defined in the classical sense and are understood by using the para-controlled distribution method. By a new characterization of weighted Hölder spaces and Zvonkin's transformation, some new a priori estimates, and the global well-posedness for singular HJB equations are proved. As applications, by using PDE arguments and para-controlled distributions, the authors also investigate the global well-posedness in polynomial weighted Hölder spaces for KPZ type equations on the real line, as well as modified KPZ equations for which the Cole-Hopf transformation is not applicable.
Reviewer: Jing Cui (Wuhu)Constant step stochastic approximations involving differential inclusions: stability, long-run convergence and applicationshttps://zbmath.org/1500.600402023-01-20T17:58:23.823708Z"Bianchi, Pascal"https://zbmath.org/authors/?q=ai:bianchi.pascal"Hachem, Walid"https://zbmath.org/authors/?q=ai:hachem.walid"Salim, Adil"https://zbmath.org/authors/?q=ai:salim.adilSummary: We consider a Markov chain \((x_n)\) whose kernel is indexed by a scaling parameter \(\gamma > 0\), referred to as the step size. The aim is to analyze the behaviour of the Markov chain in the doubly asymptotic regime where \(n \to \infty\) then \(\gamma \to \infty\). First, under mild assumptions on the so-called drift of the Markov chain, we show that the interpolated process converges narrowly to the solutions of a Differential Inclusion (DI) involving an upper semicontinuous set-valued map with closed and convex values. Second, we provide verifiable conditions which ensure the stability of the iterates. Third, by putting the above results together, we establish the long run convergence of the iterates as \(\gamma \to \infty\), to the Birkhoff center of the DI. The ergodic behaviour of the iterates is also provided. Application examples are investigated. We apply our findings to (1) the problem of nonconvex proximal stochastic optimization and (2) a fluid model of parallel queues.Cut-off phenomenon for the \(ax+b\) Markov chain over a finite fieldhttps://zbmath.org/1500.600412023-01-20T17:58:23.823708Z"Breuillard, Emmanuel"https://zbmath.org/authors/?q=ai:breuillard.emmanuel"Varjú, Péter P."https://zbmath.org/authors/?q=ai:varju.peter-palSummary: We study the Markov chain \(x_{n+1}=ax_n+b_n\) on a finite field \({\mathbb{F}}_p\), where \(a \in{\mathbb{F}}_p^{\times}\) is fixed and \(b_n\) are independent and identically distributed random variables in \({\mathbb{F}}_p\). Conditionally on the Riemann hypothesis for all Dedekind zeta functions, we show that the chain exhibits a cut-off phenomenon for most primes \(p\) and most values of \(a \in{\mathbb{F}}_p^\times \). We also obtain weaker, but unconditional, upper bounds for the mixing time.Cutoff for the asymmetric riffle shufflehttps://zbmath.org/1500.600422023-01-20T17:58:23.823708Z"Sellke, Mark"https://zbmath.org/authors/?q=ai:sellke.markSummary: In the Gilbert-Shannon-Reeds shuffle, a deck of \(N\) cards is cut into two approximately equal parts which are riffled together uniformly at random. \textit{D. Bayer} and \textit{P. Diaconis} [Ann. Appl. Probab. 2, No. 2, 294--313 (1992; Zbl 0757.60003)] famously showed that this Markov chain undergoes cutoff in total variation after \(\frac{3\log (N)}{2\log (2)}\) shuffles. We establish cutoff for the more general \textit{asymmetric} riffle shuffles in which one cuts the deck into differently sized parts. The value of the cutoff point confirms a conjecture of \textit{S. P. Lalley} from 2000 [Ann. Appl. Probab. 10, No. 4, 1302--1321 (2000; Zbl 1073.60535)]. Some appealing consequences are that asymmetry always slows mixing and that total variation mixing is strictly faster than separation and \(L^{\infty}\) mixing.A new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chainshttps://zbmath.org/1500.600432023-01-20T17:58:23.823708Z"Shchegolev, Aleksandr A."https://zbmath.org/authors/?q=ai:shchegolev.aleksandr-aSummary: In the paper, we study a new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains based on the Markov-Dobrushin condition. This result generalizes the convergence estimates for any positive number of transition steps. An example of a class of processes is provided to point that such estimates considering several transition steps may be applicable when one transition can not guarantee any convergence. Moreover, a better estimate can be obtained for a higher number of transitions steps. A law of large numbers is presented for a class of ergodic nonlinear Markov chains with finite state space that may serve as a basis for nonparametric estimation and other statistics.Approximations of piecewise deterministic Markov processes and their convergence propertieshttps://zbmath.org/1500.600442023-01-20T17:58:23.823708Z"Bertazzi, Andrea"https://zbmath.org/authors/?q=ai:bertazzi.andrea"Bierkens, Joris"https://zbmath.org/authors/?q=ai:bierkens.joris"Dobson, Paul"https://zbmath.org/authors/?q=ai:dobson.paul-wSummary: Piecewise deterministic Markov processes (PDMPs) are a class of stochastic processes with applications in several fields of applied mathematics spanning from mathematical modelling of physical phenomena to computational methods. A PDMP is specified by three characteristic quantities: the deterministic motion, the law of the random event times, and the jump kernels. The applicability of PDMPs to real world scenarios is currently limited by the fact that these processes can be simulated only when these three characteristics of the process can be simulated exactly. In order to overcome this problem, we introduce discretisation schemes for PDMPs which make their approximate simulation possible. In particular, we design both first order and higher order schemes that rely on approximations of one or more of the three characteristics. For the proposed approximation schemes we study both pathwise convergence to the continuous PDMP as the step size converges to zero and convergence in law to the invariant measure of the PDMP in the long time limit. Moreover, we apply our theoretical results to several PDMPs that arise from the computational statistics and mathematical biology literature.Geometric ergodicity of the multivariate COGARCH(1,1) processhttps://zbmath.org/1500.600452023-01-20T17:58:23.823708Z"Stelzer, Robert"https://zbmath.org/authors/?q=ai:stelzer.robert"Vestweber, Johanna"https://zbmath.org/authors/?q=ai:vestweber.johannaThe authors deduce, under the assumption of irreducibility sufficient conditions for the uniqueness of the stationary distribution, the convergence to it with an exponential rate and some finite \(p\)-moment of the stationary distribution of the MUCOGARCH (see [\textit{R. Stelzer}, Bernoulli 16, No. 1, 80--115 (2010; Zbl 1200.62110)], which is an extension of classical general autoregressive conditionally heteroscedastic (GARCH) time series models introduced by \textit{T. Bollerslev} [J. Econom. 31, 307--327 (1986; Zbl 0616.62119)]) volatility process \(Y\). They show this using the theory of Markov process, see, e.g., [\textit{D. Down} et al., Ann. Probab. 23, No. 4, 1671--1691 (1995; Zbl 0852.60075); \textit{S. P. Meyn} and \textit{R. L. Tweedie}, Adv. Appl. Probab. 25, No. 3, 487--517 (1993; Zbl 0781.60052)].
Reviewer: Romeo Negrea (Timişoara)On metastabilityhttps://zbmath.org/1500.600462023-01-20T17:58:23.823708Z"Miclo, Laurent"https://zbmath.org/authors/?q=ai:miclo.laurentThe goal of this work is to introduce a quantity which measures the metastability of an absorbed Markov chain on a finite set, without referring to the asymptotic behaviour of some parameter such as a temperature. This quantity is presented as a spectral criterion, in the sense that it is defined from the spectrum of sub-Markovian chains which are obtained from the initial one by adding a regeneration mechanism according to the quasi-stationary state and then by removing one point in the boundary of the chain (i.e., the points at which the absorbing rate is positive). The metastability is quantified in terms of the difference between the law of the absorbing time and an exponential law. The first main result is that the distance between the repartition functions of these two laws is bounded uniformly in time and in the initial condition by the spectral quantity times the absorbing rate. A similar control is proven for the joint law of the exit time and and the exit position (where metastability means that they are approximately independent, with laws independent from the initial condition). As a consequence, a metastable behaviour is observed as soon as the spectral quantity is small with respect to the inverse of the absorbing rate. The results are tested for toy models (two or three states) in a low temperature regime.
Reviewer: Pierre Monmarché (Paris)Uniform in time propagation of chaos for a Moran modelhttps://zbmath.org/1500.600472023-01-20T17:58:23.823708Z"Cloez, Bertrand"https://zbmath.org/authors/?q=ai:cloez.bertrand"Corujo, Josué"https://zbmath.org/authors/?q=ai:corujo.josue-m|corujo.josueSummary: This article studies the limit of the empirical distribution induced by a mutation-selection multi-allelic Moran model. Our results include a uniform in time bound for the propagation of chaos in \(\mathbb{L}^p\) of order \(\sqrt{N}\), and the proof of the asymptotic normality with zero mean and explicit variance, when the number of individuals tend towards infinity, for the approximation error between the empirical distribution and its limit. Additionally, we explore the interpretation of this Moran model as a particle process whose empirical probability measure approximates a quasi-stationary distribution, in the same spirit as the Fleming-Viot particle systems.How long does the surplus stay close to its historical high?https://zbmath.org/1500.600482023-01-20T17:58:23.823708Z"Li, Bo"https://zbmath.org/authors/?q=ai:li.bo.10"Hua, Yun"https://zbmath.org/authors/?q=ai:hua.yun"Zhou, Xiaowen"https://zbmath.org/authors/?q=ai:zhou.xiaowenIn their paper, the authors derive the Laplace transforms of the weighted occupation times for a spectrally negative Lévy surplus process to spend below its running maximum up to the first exit times. The authors express their results in terms of generalized scale functions. The authors also note that, for step weight functions, the Laplace transforms can be further expressed in terms of scale functions.
Reviewer: Pavel Gapeev (London)Asymptotic behavior of spectral functions for Schrödinger forms with signed measureshttps://zbmath.org/1500.600492023-01-20T17:58:23.823708Z"Wada, Masaki"https://zbmath.org/authors/?q=ai:wada.masakiSummary: Let \(\{X_t\}_{t \ge 0}\) be the rotationally invariant \(\alpha\)-stable process and define the Schrödinger forms by two methods. In one method, the perturbation is given by \(-(\mu_0 + \lambda \nu) \; (\lambda \ge 0)\), where both \(\mu_0\) and \(\nu\) are positive and \(\mu_0\) is critical. In the other method, the perturbation is given by \(\lambda \mu \; (\lambda \in \mathbb{R})\), where \(\mu\) is a critical signed measure. In this paper we consider the asymptotic behavior of the spectral functions defined from these Schrödinger forms. The results are consistent with the differentiability of the spectral functions given in [\textit{Y. Nishimori}, Tôhoku Math. J. (2) 65, No. 4, 467--494 (2013; Zbl 1294.60049); \textit{M. Takeda} and \textit{K. Tsuchida}, Trans. Am. Math. Soc. 359, No. 8, 4031--4054 (2007; Zbl 1112.60058)].
For the entire collection see [Zbl 1493.11005].Elementary symmetric polynomials and martingales for Heckman-Opdam processeshttps://zbmath.org/1500.600502023-01-20T17:58:23.823708Z"Rösler, Margit"https://zbmath.org/authors/?q=ai:rosler.margit"Voit, Michael"https://zbmath.org/authors/?q=ai:voit.michaelSummary: We consider the generators \(L_k\) of Heckman-Opdam diffusion processes in the compact and non-compact case in \(N\) dimensions for root systems of type \(A\) and \(B\), with a multiplicity function of the form \(k=\kappa k_0\) with some fixed value \(k_0\) and a varying constant \(\kappa\in[0,\infty[\). Using elementary symmetric functions, we present polynomials which are simultaneous eigenfunctions of the \(L_k\) for all \(\kappa \in]0,\infty[\). This leads to martingales associated with the Heckman-Opdam diffusions \((X_{t,1},\dots,X_{t,N})_{t\geq 0}\). As our results extend to the freezing case \(\kappa=\infty\) with a deterministic limit after some renormalization, we find formulas for the expectations \(\mathbb{E}(\prod_{j=1}^N(y-X_{t,j}))\), \(y\in\mathbb{C}\).
For the entire collection see [Zbl 1496.17001].Bayesian estimation of incompletely observed diffusionshttps://zbmath.org/1500.600512023-01-20T17:58:23.823708Z"van der Meulen, Frank"https://zbmath.org/authors/?q=ai:van-der-meulen.frank-h"Schauer, Moritz"https://zbmath.org/authors/?q=ai:schauer.moritzSummary: We present a general framework for Bayesian estimation of incompletely observed multivariate diffusion processes. Observations are assumed to be discrete in time, noisy and incomplete. We assume the drift and diffusion coefficient depend on an unknown parameter. A data-augmentation algorithm for drawing from the posterior distribution is presented which is based on simulating diffusion bridges conditional on a noisy incomplete observation at an intermediate time. The dynamics of such filtered bridges are derived and it is shown how these can be simulated using a generalised version of the guided proposals introduced in
[\textit{M. Schauer} et al., Bernoulli 23, No. 4A, 2917--2950 (2017; Zbl 1415.65022)].Diffusion approximation for a simple kinetic model with asymmetric interfacehttps://zbmath.org/1500.600522023-01-20T17:58:23.823708Z"Bobrowski, Adam"https://zbmath.org/authors/?q=ai:bobrowski.adam"Komorowski, Tomasz"https://zbmath.org/authors/?q=ai:komorowski.tomaszSummary: We study a diffusion approximation for a model of stochastic motion of a particle in one spatial dimension. The velocity of the particle is constant but the direction of the motion undergoes random changes with a Poisson clock. Moreover, the particle interacts with an interface in such a way that it can randomly be reflected, transmitted, or killed, and the corresponding probabilities depend on whether the particle arrives at the interface from the left or right. We prove that the limit process is a \textit{minimal Brownian motion}, if the probability of killing is positive. In the case of no killing, the limit is a \textit{skew Brownian motion}. Moreover, we construct a cosine family related to the skew Brownian motion and provide a new derivation of transition probability densities for this process.The Brown measure of the free multiplicative Brownian motionhttps://zbmath.org/1500.600532023-01-20T17:58:23.823708Z"Driver, Bruce K."https://zbmath.org/authors/?q=ai:driver.bruce-k"Hall, Brian"https://zbmath.org/authors/?q=ai:hall.brian-c"Kemp, Todd"https://zbmath.org/authors/?q=ai:kemp.todd-aSummary: The free multiplicative Brownian motion \(b_t\) is the large-\(N\) limit of the Brownian motion on \(\mathsf{GL}(N;\mathbb{C}),\) in the sense of \(*\)-distributions. The natural candidate for the large-\(N\) limit of the empirical distribution of eigenvalues is thus the Brown measure of \(b_t \). In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region \(\Sigma_t\) that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density \(W_t\) on \(\overline{\Sigma}_t,\) which is strictly positive and real analytic on \(\Sigma_t\). This density has a simple form in polar coordinates:
\[
W_t(r,\theta)=\frac{1}{r^2}w_t(\theta),
\] where \(w_t\) is an analytic function determined by the geometry of the region \(\Sigma_t\). We show also that the spectral measure of free unitary Brownian motion \(u_t\) is a ``shadow'' of the Brown measure of \(b_t\), precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.Existence and Hölder continuity conditions for self-intersection local time of Rosenblatt processhttps://zbmath.org/1500.600542023-01-20T17:58:23.823708Z"Yu, Qian"https://zbmath.org/authors/?q=ai:yu.qian"Shen, Guangjun"https://zbmath.org/authors/?q=ai:shen.guangjun"Yin, Xiuwei"https://zbmath.org/authors/?q=ai:yin.xiuweiThe Rosenblatt process is considered in the paper. The existence and Hölder continuity conditions for the self-intersection local time and collision local time are investigated.
Reviewer: Rózsa Horváth-Bokor (Budakalász)Convergence rate for a class of supercritical superprocesseshttps://zbmath.org/1500.600552023-01-20T17:58:23.823708Z"Liu, Rongli"https://zbmath.org/authors/?q=ai:liu.rongli"Ren, Yan-Xia"https://zbmath.org/authors/?q=ai:ren.yanxia"Song, Renming"https://zbmath.org/authors/?q=ai:song.renmingSummary: Suppose \(X=\{X_t,t\geq 0\}\) is a supercritical superprocess. Let \(\phi\) be the non-negative eigenfunction of the mean semigroup of \(X\) corresponding to the principal eigenvalue \(\lambda > 0\). Then \(M_t(\phi)=e^{-\lambda t}\langle\phi,X_t\rangle,t\geq 0\), is a non-negative martingale with almost sure limit \(M_\infty(\phi)\). In this paper we study the rate at which \(M_t(\phi)-M_\infty(\phi)\) converges to \(0\) as \(t\to\infty\) when the process may not have finite variance. Under some conditions on the mean semigroup, we provide sufficient and necessary conditions for the rate in the almost sure sense. Some results on the convergence rate in \(L^p\) with \(p\in(1,2)\) are also obtained.Brownian bees in the infinite swarm limithttps://zbmath.org/1500.600562023-01-20T17:58:23.823708Z"Berestycki, Julien"https://zbmath.org/authors/?q=ai:berestycki.julien"Brunet, Éric"https://zbmath.org/authors/?q=ai:brunet.eric"Nolen, James"https://zbmath.org/authors/?q=ai:nolen.james|nolen.james-h"Penington, Sarah"https://zbmath.org/authors/?q=ai:penington.sarahSummary: The \textit{Brownian bees} model is a branching particle system with spatial selection. It is a system of \(N\) particles which move as independent Brownian motions in \(\mathbb{R}^d\) and independently branch at rate 1, and, crucially, at each branching event, the particle which is the furthest away from the origin is removed to keep the population size constant. In the present work we prove that, as \(N\to \infty\), the behaviour of the particle system is well approximated by the solution of a free boundary problem (which is the subject of a companion paper [the authors, Trans. Am. Math. Soc. 374, No. 9, 6269--6329 (2021; Zbl 1475.35420)]), the \textit{hydrodynamic limit} of the system. We then show that for this model the so-called \textit{selection principle} holds; that is, that as \(N\to \infty\), the equilibrium density of the particle system converges to the steady-state solution of the free boundary problem.Quenched local central limit theorem for random walks in a time-dependent balanced random environmenthttps://zbmath.org/1500.600572023-01-20T17:58:23.823708Z"Deuschel, Jean-Dominique"https://zbmath.org/authors/?q=ai:deuschel.jean-dominique"Guo, Xiaoqin"https://zbmath.org/authors/?q=ai:guo.xiaoqinThe authors consider a random walk in a balanced uniformly-elliptic time-dependent random environment on \(\mathbb{Z}^{d}\), \(d \geq 2\). A quenched local central limit theorem is proved for continuous-time random walks, in a uniformly-elliptic time-dependent balanced random environment which is ergodic under space-time shifts. Moreover, Gaussian upper and lower bounds for quenched and moment estimates of the transition probabilities and asymptotics of the discrete Green's function are obtained.
Reviewer: Utkir A. Rozikov (Tashkent)Polynomial ballisticity conditions and invariance principle for random walks in strong mixing environmentshttps://zbmath.org/1500.600582023-01-20T17:58:23.823708Z"Guerra, Enrique"https://zbmath.org/authors/?q=ai:guerra.enrique"Valle, Glauco"https://zbmath.org/authors/?q=ai:valle-da-silva-coelho.glauco"Vares, Maria Eulália"https://zbmath.org/authors/?q=ai:vares.maria-eulaliaThe nearest-neighbor \(d\)-dimensional random walk in a random environment (RWRE) \((X_n)_{n\geq 0}\) is considered on the lattice \(\mathbb{Z}^d\) (\(d \geq 2\)) in more general than i.i.d., the so-called mixing setting. Let \(P_0\) be the averaged or annealed law of a RWRE with \(X_0=0\). Given a vector \(\ell \in \mathbb{S}^{d-1}\), a RWRE is called a ballistic in the direction \(\ell\) if \(\liminf_{n \to \infty} n^{-1} (X \cdot \ell) >0\) (\(P_0\)-a.s.). In this article, along with asymptotic criteria an effective polynomial condition of ballisticity is investigated. The effective character of the criterion means that it can be checked by inspection on a finite box.
The first result is the proof of the RWRE conjecture of Sznitman in considering a mixing setting: under some sufficient conditions and a strong mixing condition the following statements are equivalent: i) \((T^\gamma)|\ell\) holds; ii) \((T')|\ell\) holds; iii) \((P^*_J)|\ell\) holds; iv) \((EC)|\ell\) holds. The second result is the annealed invariance principle: under some sufficient conditions and a strong mixing condition there exist a deterministic non-degenerate covariance matrix \(R\) and a deterministic vector \(v\) with \(v \cdot \ell > 0\) such that, setting \(S_n(t) := ( X_{[nt]} - vt)n^{-1/2}\), the process \(S_n(t)\) converges in law under \(P_0\) to a standard Brownian motion with covariance matrix \(R\).
Reviewer: Alex V. Kolnogorov (Novgorod)An invariance principle for ergodic scale-free random environmentshttps://zbmath.org/1500.600592023-01-20T17:58:23.823708Z"Gwynne, Ewain"https://zbmath.org/authors/?q=ai:gwynne.ewain"Miller, Jason"https://zbmath.org/authors/?q=ai:miller.jason-p"Sheffield, Scott"https://zbmath.org/authors/?q=ai:sheffield.scottIn the paper the authors consider random planar lattices whose laws are ``translation invariant modulo scaling'' in a specially defined sense, which is not necessarily translation invariant in the usual sense. The goal of the paper is to prove that simple random walks on certain random planar lattices have Brownian motion as a scaling limit. The main results of the paper are variants of the following: if a random embedded lattice is ergodic modulo scaling and satisfies an appropriate ``finite specific Dirichlet energy'' condition, then a simple random walk on embedded lattice scales to a Brownian motion modulo time parametrization in the quenched sense.
Reviewer: Utkir A. Rozikov (Tashkent)Optimal signal detection in some spiked random matrix models: likelihood ratio tests and linear spectral statisticshttps://zbmath.org/1500.620012023-01-20T17:58:23.823708Z"Banerjee, Debapratim"https://zbmath.org/authors/?q=ai:banerjee.debapratim"Ma, Zongming"https://zbmath.org/authors/?q=ai:ma.zongmingSummary: We study signal detection by likelihood ratio tests in a number of spiked random matrix models, including but not limited to Gaussian mixtures and spiked Wishart covariance matrices. We work directly with multi-spiked cases in these models and with flexible priors on signal components that allow dependence across spikes. We derive asymptotic normality for the log-likelihood ratios when the signal-to-noise ratios are below certain bounds. In addition, the log-likelihood ratios can be asymptotically decomposed as weighted sums of a collection of statistics which we call bipartite signed cycles. Based on this decomposition, we show that below the bounds we could always achieve the asymptotically optimal powers of likelihood ratio tests via tests based on linear spectral statistics which have polynomial time complexity.Exact minimax risk for linear least squares, and the lower tail of sample covariance matriceshttps://zbmath.org/1500.620022023-01-20T17:58:23.823708Z"Mourtada, Jaouad"https://zbmath.org/authors/?q=ai:mourtada.jaouadSummary: We consider random-design linear prediction and related questions on the lower tail of random matrices. It is known that, under boundedness constraints, the minimax risk is of order \(d/n\) in dimension \(d\) with \(n\) samples. Here, we study the minimax expected excess risk over the full linear class, depending on the distribution of covariates. First, the least squares estimator is exactly minimax optimal in the well-specified case, for every distribution of covariates. We express the minimax risk in terms of the distribution of statistical leverage scores of individual samples, and deduce a minimax lower bound of \(d/(n-d+1)\) for any covariate distribution, nearly matching the risk for Gaussian design. We then obtain sharp nonasymptotic upper bounds for covariates that satisfy a ``small ball''-type regularity condition in both well-specified and misspecified cases.
Our main technical contribution is the study of the lower tail of the smallest singular value of empirical covariance matrices at small values. We establish a lower bound on this lower tail, valid for any distribution in dimension \(d\ge 2\), together with a matching upper bound under a necessary regularity condition. Our proof relies on the PAC-Bayes technique for controlling empirical processes, and extends an analysis of Oliveira devoted to a different part of the lower tail.A numerical scheme for stochastic differential equations with distributional drifthttps://zbmath.org/1500.650012023-01-20T17:58:23.823708Z"De Angelis, Tiziano"https://zbmath.org/authors/?q=ai:de-angelis.tiziano"Germain, Maximilien"https://zbmath.org/authors/?q=ai:germain.maximilien"Issoglio, Elena"https://zbmath.org/authors/?q=ai:issoglio.elenaSummary: In this paper we introduce a scheme for the numerical solution of one-dimensional stochastic differential equations (SDEs) whose drift belongs to a fractional Sobolev space of negative regularity (a subspace of Schwartz distributions). We obtain a convergence rate in a suitable \(L^1\)-norm and, as a by-product, a convergence rate for a numerical scheme applied to SDEs with drift in \(L^p\)-spaces with \(p\in(1,\infty)\).Quasi-Toeplitz trigonometric transform splitting methods for spatial fractional diffusion equationshttps://zbmath.org/1500.650472023-01-20T17:58:23.823708Z"Shao, Xin-Hui"https://zbmath.org/authors/?q=ai:shao.xinhui"Li, Yu-Han"https://zbmath.org/authors/?q=ai:li.yuhan"Shen, Hai-Long"https://zbmath.org/authors/?q=ai:shen.hailongSummary: The random walk model describing the super-diffusion competition phenomenon of particles can derive the spatial fractional diffusion equation. For irregular diffusion and super-diffusion phenomena, the use of such equation can obtain more accurate and realistic results, so it has a wide application background in practice. The implicit finite-difference method derived from the shifted Grünwald scheme is used to discretize the spatial fractional diffusion equation. The coefficient matrix of discrete system is in the form of the sum of a diagonal matrix and a Toeplitz matrix. In this paper, a preconditioner is constructed, which transforms the coefficient matrix into the form of an identity matrix plus a diagonal matrix multiplied by Toeplitz matrix. On this basis, a new quasi-Toeplitz trigonometric transform splitting iteration format (abbreviated as QTTTS method) is proposed. We theoretically verify the unconditional convergence of the new method, and obtain the effective optimal form of the iteration parameter. Finally, numerical simulation experiments also demonstrate the accurateness and efficiency of the new method.Numerical identification of nonlocal potentials in aggregationhttps://zbmath.org/1500.650542023-01-20T17:58:23.823708Z"He, Yuchen"https://zbmath.org/authors/?q=ai:he.yuchen"Kang, Sung Ha"https://zbmath.org/authors/?q=ai:kang.sung-ha"Liao, Wenjing"https://zbmath.org/authors/?q=ai:liao.wenjing"Liu, Hao"https://zbmath.org/authors/?q=ai:liu.hao.2|liu.hao|liu.hao.1"Liu, Yingjie"https://zbmath.org/authors/?q=ai:liu.yingjieSummary: Aggregation equations are broadly used to model population dynamics with nonlocal interactions, characterized by a potential in the equation. This paper considers the inverse problem of identifying the potential from a single noisy spatial-temporal process. The identification is challenging in the presence of noise due to the instability of numerical differentiation. We propose a robust model-based technique to identify the potential by minimizing a regularized data fidelity term, and regularization is taken as the total variation and the squared Laplacian. A split Bregman method is used to solve the regularized optimization problem. Our method is robust to noise by utilizing a Successively Denoised Differentiation technique. We consider additional constraints such as compact support and symmetry constraints to enhance the performance further. We also apply this method to identify time-varying potentials and identify the interaction kernel in an agent-based system. Various numerical examples in one and two dimensions are included to verify the effectiveness and robustness of the proposed method.Exponentially convergent method for the approximation of a differential equation with fractional derivative and unbounded operator coefficient in a Banach spacehttps://zbmath.org/1500.650902023-01-20T17:58:23.823708Z"Vasylyk, V. B."https://zbmath.org/authors/?q=ai:vasylyk.v-b"Gavrilyuk, I. P."https://zbmath.org/authors/?q=ai:gavrilyuk.ivan-p"Makarov, V. L."https://zbmath.org/authors/?q=ai:makarov.volodymyr-lSummary: We propose and analyze an exponentially convergent numerical method for solving differential equations with right-hand fractional Riemann-Liouville derivative and unbounded operator coefficient in Banach spaces. We use a representation of the solution in the form of the Danford-Cauchy integral on a hyperbola that covers the spectrum of the operator coefficient with subsequent application of the exponentially convergent quadrature. To this end, we choose the parameters of the hyperbola in order to guarantee the possibility of analytic extension of the integrand in a strip containing the real axis and then apply the Sinc-quadrature. We prove the exponential accuracy of the method and present a numerical example that confirms the obtained \textit{a priori} estimate.Construction of a class of copula using the finite difference methodhttps://zbmath.org/1500.650912023-01-20T17:58:23.823708Z"Bagré, Remi Guillaume"https://zbmath.org/authors/?q=ai:bagre.remi-guillaume"Béré, Frédéric"https://zbmath.org/authors/?q=ai:bere.frederic"Loyara, Vini Yves Bernadin"https://zbmath.org/authors/?q=ai:loyara.vini-yves-bernadinSummary: The definition of a copula function and the study of its properties are at the same time not obvious tasks, as there is no general method for constructing them. In this paper, we present a method that allows us to obtain a class of copula as a solution to a boundary value problem. For this, we use the finite difference method which is a common technique for finding approximate solutions of partial differential equations which consists in solving a system of relations (numerical scheme) linking the values of the unknown functions at certain points sufficiently close to each other.The gambler's ruin problem and quantum measurementhttps://zbmath.org/1500.810032023-01-20T17:58:23.823708Z"Debbasch, Fabrice"https://zbmath.org/authors/?q=ai:debbasch.fabriceSummary: The dynamics of a single microscopic or mesoscopic non quantum system interacting with a macroscopic environment is generally stochastic. In the same way, the reduced density operator of a single quantum system interacting with a macroscopic environment is \textit{a priori} a stochastic variable, and decoherence describes only the average dynamics of this variable, not its fluctuations. It is shown that a general unbiased quantum measurement can be reformulated as a gambler's ruin problem where the game is a martingale. Born's rule then appears as a direct consequence of the optional stopping theorem for martingales. Explicit computations are worked out in detail on a specific simple example.
For the entire collection see [Zbl 1466.81003].Classification of classical non-Gaussian noises with respect to their detrimental effects on the evolution of entanglement using a system of three-qubit as probehttps://zbmath.org/1500.810082023-01-20T17:58:23.823708Z"Kenfack, Lionel Tenemeza"https://zbmath.org/authors/?q=ai:kenfack.lionel-tenemeza"Tchoffo, Martin"https://zbmath.org/authors/?q=ai:tchoffo.martin"Fai, Lukong Cornelius"https://zbmath.org/authors/?q=ai:fai.lukong-corneliusSummary: A system of three non-interacting qubits is used as a quantum probe to classify three classical non-Gaussian noises namely, the static noise (SN), colored noise (pink and brown spectrum) and random telegraph noise (RTN), according to their detrimental effects on the evolution of entanglement of the latter. The probe system is initially prepared in the GHZ state and coupled to the noises in independent environments. Seven configurations for the qubit-noise coupling (QNC) are considered. To estimate the destructive influence of each kind of noise, the tripartite negativity is employed to compare the evolution of entanglement in these QNC configurations to each other with the same noise parameters. It is shown that the evolution of entanglement is drastically impacted by the QNC configuration considered as well as the properties of the environmental noises and that the SN is more detrimental to the survival of entanglement than the RTN and colored noise, regardless of the Markov or non-Markov character of the RTN and the spectrum of the colored noise. On the other hand, it is shown that pink noise is more fatal to the system than the RTN and that the situation is totally reversed in the case of brown noise. Finally, it is demonstrated that these noises, in descending order of destructive influence, can be classified as follows: SN \(>\) pink noise \(>\text{RTN}>\) brown noise.Complex time method for quantum dynamics when an exceptional point is encircled in the parameter spacehttps://zbmath.org/1500.810332023-01-20T17:58:23.823708Z"Kaprálová-Žďánská, Petra Ruth"https://zbmath.org/authors/?q=ai:kapralova-zdanska.petra-ruthSummary: We revisit the complex time method for the application to quantum dynamics as an exceptional point is encircled in the parameter space of the Hamiltonian. The basic idea of the complex time method is using complex contour integration to perform the first-order adiabatic perturbation integral. In this way, the quantum dynamical problem is transformed to a study of singularities in the complex time plane -- transition points -- which represent complex degeneracies of the adiabatic Hamiltonian as the time-dependent parameters defining the encircling contour are analytically continued to complex plane. As an underlying illustration of the approach we discuss a switch between Rabi oscillations and rapid adiabatic passage which occurs upon the encircling of an exceptional point in a special time-symmetric case.A generalized quantum optical scheme for implementing open quantum walkshttps://zbmath.org/1500.810552023-01-20T17:58:23.823708Z"Zungu, Ayanda Romanis"https://zbmath.org/authors/?q=ai:zungu.ayanda-romanis"Sinayskiy, Iiya"https://zbmath.org/authors/?q=ai:sinayskiy.iiya"Petruccione, Francesco"https://zbmath.org/authors/?q=ai:petruccione.francescoSummary: Open quantum walks (OQWs) are a new type of quantum walks which are entirely driven by the dissipative interaction with external environments and are formulated as completely positive trace-preserving maps on graphs. A generalized quantum optical scheme for implementing OQWs that includes non-zero temperature of the environment is suggested. In the proposed quantum optical scheme, a two-level atom plays the role of the ``walker'', and the Fock states of the cavity mode correspond to the lattice sites for the ``walker''. Using the small unitary rotations approach the effective dynamics of the system is shown to be an OQW. For the chosen set of parameters, an increase in the temperature of the environment causes the system to reach the asymptotic distribution much faster compared to the scheme proposed earlier where the temperature of the environment is zero. For this case the asymptotic distribution is given by a steady Gaussian distribution.
For the entire collection see [Zbl 1466.81003].A random matrix model with non-pairwise contracted indiceshttps://zbmath.org/1500.810612023-01-20T17:58:23.823708Z"Lionni, Luca"https://zbmath.org/authors/?q=ai:lionni.luca"Sasakura, Naoki"https://zbmath.org/authors/?q=ai:sasakura.naokiSummary: We consider a random matrix model with both pairwise and non-pairwise contracted indices. The partition function of the matrix model is similar to that appearing in some replicated systems with random tensor couplings, such as the \(p\)-spin spherical model for the spin glass. We analyze the model using Feynman diagrammatic expansions, and provide an exhaustive characterization of the graphs that dominate when the dimensions of the pairwise and (or) non-pairwise contracted indices are large. We apply this to investigate the properties of the wave function of a toy model closely related to a tensor model in the Hamilton formalism, which is studied in a quantum gravity context, and obtain a result in favor of the consistency of the quantum probabilistic interpretation of this tensor model.A variational framework for the inverse Henderson problem of statistical mechanicshttps://zbmath.org/1500.820072023-01-20T17:58:23.823708Z"Frommer, Fabio"https://zbmath.org/authors/?q=ai:frommer.fabio"Hanke, Martin"https://zbmath.org/authors/?q=ai:hanke-bourgeois.martinThe inverse Henderson problem refers to the determination of the pair potential which specifies the interactions in an ensemble of classical particles in continuous space, given the density and the equilibrium pair correlation function of these particles as data. In the paper, the authors show that in the thermodynamic limit analogous connections exist between the specific relative entropy introduced by \textit{H.-O. Georgii} and \textit{H. Zessin} [Probab. Theory Relat. Fields 96, No. 2, 177--204 (1993; Zbl 0792.60024)] and a proper formulation of the inverse Monte Carlo iteration in the full space. This provides a rigorous variational framework for the inverse Henderson problem, valid within a large class of pair potentials.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)Mirror symmetry of height-periodic gradient Gibbs measures of an SOS model on Cayley treeshttps://zbmath.org/1500.820082023-01-20T17:58:23.823708Z"Rozikov, U. A."https://zbmath.org/authors/?q=ai:rozikov.utkir-aFor the solid-on-solid model with spin values from the set of all integers on a Cayley tree, the author specifies a gradient Gibbs measures (GGMs). Such a measure corresponds to a boundary law (which is an infinite-dimensional vector-valued function defined on vertices of the Cayley tree) satisfying an infinite system of functional equations. In previous papers [\textit{F. H. Haydarov} and \textit{U. A. Rozikov}, Rep. Math. Phys. 90, No. 1, 81--101 (2022; Zbl 1495.82005); \textit{F. Henning} et al., Electron. J. Probab. 24, Paper No. 104, 23 p. (2019; Zbl 1431.82014)] some 4-periodic boundary laws were found. In this paper the author considers new boundary laws which are independent from vertices of the Cayley tree and (as an infinite-dimensional vector) have periodic, (non-) mirror-symmetric coordinates. Namely, the particular class of height-periodic boundary laws of period \(q \leq 5\) is studied, where solutions are classified by their period and (two-) mirror-symmetry.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)Irreversibility and the arrow of timehttps://zbmath.org/1500.820092023-01-20T17:58:23.823708Z"Fröhlich, Jürg"https://zbmath.org/authors/?q=ai:frohlich.jurg-martinSummary: Within the general formalism of quantum theory irreversibility and the arrow of time in the evolution of various physical systems are studied. Irreversible behavior often manifests itself in the guise of ``entropy production.''This motivates me to begin this chapter with a brief review of quantum-mechanical entropy, a subject that Elliott Lieb has made outstanding contributions to, followed by an enumeration of examples of irreversible behavior and of an arrow of time analyzed in later sections. Subsequently, a derivation of the laws of thermodynamics from (quantum) statistical mechanics, and, in particular, of the Second Law of thermodynamics, in the forms given to it by Clausius and Carnot, is presented. In a third part, results on diffusive (Brownian) motion of a quantum particle interacting with a quasi-free quantum-mechanical heat bath are reviewed. This is followed by an outline of a theory of friction by emission of Cherenkov radiation of sound waves in a system consisting of a particle moving through a Bose-Einstein condensate and interacting with it. In what may be the most important section of this chapter, the fundamental arrow of timeinherent in Quantum Mechanics is discussed.
For the entire collection see [Zbl 1491.46002].A fluid-diffusion-hybrid limiting approximation for priority systems with fast and slow customershttps://zbmath.org/1500.900142023-01-20T17:58:23.823708Z"Yu, Lun"https://zbmath.org/authors/?q=ai:yu.lun"Iravani, Seyed"https://zbmath.org/authors/?q=ai:iravani.seyed-m-r"Perry, Ohad"https://zbmath.org/authors/?q=ai:perry.ohadSummary: We consider a large service system with two customer classes that are distinguished by their urgency and service requirements. In particular, one of the customer classes is considered urgent, and is therefore prioritized over the other class; further, the average service time of customers from the urgent class is significantly larger than that of the nonurgent class. We therefore refer to the urgent class as ``slow,'' and to the nonurgent class as ``fast.'' Due to the complexity and intractability of the system's dynamics, our goal is to develop and analyze an asymptotic approximation, which captures the prevalent fact that, in practice, customers from both classes are likely to experience delays before entering service. However, under existing many-server limiting regimes, only two of the following options can be captured in the limit: (i) either the customers from the prioritized (slow) customer class do not wait at all, or (ii) the fast-class customers do not receive any service. We therefore propose a novel \textit{Fluid-Diffusion Hybrid} (FDH) many-server asymptotic regime, under which the queue of the slow class behaves like a diffusion limit, whereas the queue of the fast class evolves as a (random) fluid limit that is driven by the diffusion process. That FDH limit is achieved by assuming that the service rate of the fast class scales with the system's size, whereas the service rate of the slow class is kept fixed. Numerical examples demonstrate that our FDH limit is accurate when the difference between the service rates of the two classes is sufficiently large. We then employ the FDH approximation to study the costs and benefits of de-pooling the service pool, by reserving a small number of servers for the fast class. We prove that, in some cases, a two-pool structure is the asymptotically optimal system design.Stackelberg stochastic differential game with asymmetric noisy observationshttps://zbmath.org/1500.910122023-01-20T17:58:23.823708Z"Zheng, Yueyang"https://zbmath.org/authors/?q=ai:zheng.yueyang"Shi, Jingtao"https://zbmath.org/authors/?q=ai:shi.jingtaoSummary: This paper is concerned with a Stackelberg stochastic differential game with asymmetric noisy observation. In our model, the follower cannot observe the state process directly, but could observe a noisy observation process, while the leader can completely observe the state process. Open-loop Stackelberg equilibrium is considered. The follower first solve a stochastic optimal control problem with partial observation, the maximum principle and verification theorem are obtained. Then the leader turns to solve an optimal control problem for a conditional mean-field forward-backward stochastic differential equation, and both maximum principle and verification theorem are proved. A linear-quadratic Stackelberg stochastic differential game with asymmetric noisy observation is discussed to illustrate the theoretical results in this paper. With the aid of some new Riccati equations, the open-loop Stackelberg equilibrium admits its state estimate feedback representation. Finally, an application to the resource allocation and its numerical simulation are given to show the effectiveness of the proposed results.Mean field games with common noises and conditional distribution dependent FBSDEshttps://zbmath.org/1500.910142023-01-20T17:58:23.823708Z"Huang, Ziyu"https://zbmath.org/authors/?q=ai:huang.ziyu"Tang, Shanjian"https://zbmath.org/authors/?q=ai:tang.shanjianSummary: In this paper, the authors consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way. They assume that the cost function satisfies a convexity and a weak monotonicity property. They use the sufficient Pontryagin principle for optimality to transform the mean field control problem into existence and uniqueness of solution of conditional distribution dependent forward-backward stochastic differential equation (FBSDE for short). They prove the existence and uniqueness of solution of the conditional distribution dependent FBSDE when the dependence of the state on the conditional distribution is sufficiently small, or when the convexity parameter of the running cost on the control is sufficiently large. Two different methods are developed. The first method is based on a continuation of the coefficients, which is developed for FBSDE by \textit{Y. Hu} and \textit{S. Peng} [Probab. Theory Relat. Fields 103, No. 2, 273--283 (1995; Zbl 0831.60065)]. They apply the method to conditional distribution dependent FBSDE. The second method is to show the existence result on a small time interval by Banach fixed point theorem and then extend the local solution to the whole time interval.Extended Sprague-Grundy theory for locally finite games, and applications to random game-treeshttps://zbmath.org/1500.910382023-01-20T17:58:23.823708Z"Martin, James B."https://zbmath.org/authors/?q=ai:martin.james-b.1|martin.james-bSummary: The Sprague-Grundy theory for finite games without cycles was extended to general finite games by \textit{C. A. B. Smith} [J. Comb. Theory 1, 51--81 (1966; Zbl 0141.36101)] and by \textit{A. S. Fraenkel} and \textit{Y. Yesha} [J. Comb. Theory, Ser. A 43, 165--177 (1986; Zbl 0622.05030)]. We observe that the same framework used to classify finite games also covers the case of locally finite games (that is, games where any position has only finitely many options). In particular, any locally finite game is equivalent to some finite game. We then study cases where the directed graph of a game is chosen randomly and is given by the tree of a Galton-Watson branching process. Natural families of offspring distributions display a surprisingly wide range of behavior. The setting shows a nice interplay between ideas from combinatorial game theory and ideas from probability.
For the entire collection see [Zbl 1495.91007].Impact of financial crisis on economic growth: a stochastic modelhttps://zbmath.org/1500.910962023-01-20T17:58:23.823708Z"Tadmon, Calvin"https://zbmath.org/authors/?q=ai:tadmon.calvin"Tchaptchet, Eric Rostand Njike"https://zbmath.org/authors/?q=ai:njike-tchaptchet.eric-rostand(no abstract)Valuation of a DB underpin hybrid pension under a regime-switching Lévy modelhttps://zbmath.org/1500.911082023-01-20T17:58:23.823708Z"Ai, Meiqiao"https://zbmath.org/authors/?q=ai:ai.meiqiao"Zhang, Zhimin"https://zbmath.org/authors/?q=ai:zhang.zhimin.1"Zhong, Wei"https://zbmath.org/authors/?q=ai:zhong.wei|zhong.wei.1|zhong.wei.2Summary: This paper studies the valuation problem of the defined benefit (DB) underpin guarantee. We consider that the salary process follows a geometric Brownian motion, and the stochastic price index process of the funds in the defined contribution (DC) account is modeled by a regime-switching Lévy process. Under this framework, the explicit valuation formula of the DB underpin option is derived by the Fourier cosine series expansion (COS) method, and the corresponding error analysis is provided. Numerous simulation experiments are performed to illustrate the accuracy and efficiency of the proposed method. In addition, the convergence of this method and its sensitivity with respect to various model paraments are analyzed.High dimensional Markovian trading of a single stockhttps://zbmath.org/1500.911302023-01-20T17:58:23.823708Z"Elliott, Robert"https://zbmath.org/authors/?q=ai:elliott.robert-j"Madan, Dilip B."https://zbmath.org/authors/?q=ai:madan.dilip-b"Wang, King"https://zbmath.org/authors/?q=ai:wang.king-hangSummary: OU processes with long term drifts that are tempered fractional Lévy processes reduce to a \(d+1\) dimensional Markovian system when the parameter \(d\) is an integer. Markovian optimization problems are formulated for the proportion of a dollar to be invested in a risky stock following the specified dynamics. The objective evaluates the cumulated discounted returns to a dollar being invested continuously through time. Risk sensitivity is accomplished by maximizing a conservative financial valuation seen as a nonlinear expectation. Trading policies are determined by solutions of nonlinear partial integro-differential equations. The policies are evaluated on a quantized set of representative Markovian states in the higher dimensions. Gaussian process regressions are then employed to deliver general functions of the state. The nonlinear policy functions deliver good trading outcomes on simulated data. The policy functions are then applied to trading \textit{SPY} from 2008 through 2020 with good results. They are also employed to trade 874 stocks over a four year period with reasonable results. Only three policy functions trained on one year of \textit{SPY} data for 2020 are reported on. It is conjectured that a variety of functions may be trained on other data sets over other periods and selections may then be made for the functions actually traded on a particular stock at a particular time from this collection. The underlying dynamics may also be further enriched by allowing for a Markov chain of states that code changes in the parameter values for the driving Lévy process.Moments of integrated exponential Lévy processes and applications to Asian options pricinghttps://zbmath.org/1500.911322023-01-20T17:58:23.823708Z"Brignone, Riccardo"https://zbmath.org/authors/?q=ai:brignone.riccardoSummary: We find explicit formulas for the moments of the time integral of an exponential Lévy process. We consider both the cases of unconditional moments and conditional on the Lévy process level at the endpoints of the time interval. We propose a new methodology for reconstructing the unknown density of the time integral based on unconditional moments and an efficient simulation scheme based on conditional moments. These methodologies are applied for Asian option pricing, an important problem in financial literature.\( G\)-expectation approach to stochastic orderinghttps://zbmath.org/1500.911372023-01-20T17:58:23.823708Z"Ly, Sel"https://zbmath.org/authors/?q=ai:ly.sel"Privault, Nicolas"https://zbmath.org/authors/?q=ai:privault.nicolasSummary: This paper studies stochastic ordering under nonlinear expectations \(\mathcal{E}_{\mathcal{G}}\) generated by solutions of \( G \)-backward stochastic differential equations (\(G \)-BSDEs) defined on \( G \)-expectation spaces. We derive sufficient conditions for the convex, increasing convex, and monotonic \( G \)-stochastic orderings of \( G \)-diffusion processes at terminal time. Our approach relies on comparison properties for \( G \)-forward-backward stochastic differential equations (\(G\)-FBSDEs) and on relevant extensions of convexity, monotonicity and continuous dependence properties for the solutions of associated Hamilton-Jacobi-Bellman (HJB) equations. Applications of \( G \)-stochastic ordering to contingent claim superhedging price comparison under ambiguous coefficients are provided.Implied price processes anchored in statistical realizationshttps://zbmath.org/1500.911382023-01-20T17:58:23.823708Z"Madan, Dilip B."https://zbmath.org/authors/?q=ai:madan.dilip-b"Wang, King"https://zbmath.org/authors/?q=ai:wang.king-hangSummary: It is observed that statistical and risk neutral densities of compound Poisson processes are unconstrained relative to each other. Continuous processes are too constrained and generally not consistent with market data. Pure jump limit laws deliver operational models simultaneously consistent with both data sets with the additional imposition of no measure change on the arbitrarily small moves. The measure change density must have a finite Hellinger distance from unity linking the two worlds. Models are constructed using the bilateral gamma and the CGMY models for the risk neutral specification. They are linked to the physical process by measure change models. The resulting models simultaneously calibrate statistical tail probabilities and option prices. The resulting models have up to eight or ten parameters permitting the study of risk reward relations at a finer level. Rewards measured by power variations of the up and down moves are observed to value negatively(positively) the even(odd) variations of their own side with the converse holding for the opposite side.Asset price bubbles in markets with transaction costshttps://zbmath.org/1500.911412023-01-20T17:58:23.823708Z"Biagini, Francesca"https://zbmath.org/authors/?q=ai:biagini.francesca"Reitsam, Thomas"https://zbmath.org/authors/?q=ai:reitsam.thomasSummary: We study asset price bubbles in market models with proportional transaction costs \(\lambda\in (0, 1)\) and finite time horizon \(T\) in the setting of \textit{W. Schachermayer} [Lect. Notes Math. 2123, 317--331 (2014; Zbl 1390.91286)]. By following \textit{M. Herdegen} and \textit{M. Schweizer} [Int. J. Theor. Appl. Finance 19, No. 4, Article ID 1650022, 44 p. (2016; Zbl 1350.91019)], we define the fundamental value \( F\) of a risky asset \(S \) as the price of a super-replicating portfolio for a position terminating in one unit of the asset and zero cash. We then obtain a dual representation for the fundamental value by using the super-replication theorem of \textit{W. Schachermayer} [Math. Financ. Econ. 8, No. 4, 383--398 (2014; Zbl 1309.91136)]. We say that an asset price has a bubble if its fundamental value differs from the ask-price \((1+\lambda)S \). We investigate the impact of transaction costs on asset price bubbles and show that our model intrinsically includes the birth of a bubble.Stochastic modelling and simulation of PTEN regulatory networks with miRNAs and ceRNAshttps://zbmath.org/1500.920012023-01-20T17:58:23.823708Z"Carletti, Margherita"https://zbmath.org/authors/?q=ai:carletti.margherita"Spaletta, Giulia"https://zbmath.org/authors/?q=ai:spaletta.giuliaSummary: In this work, three genetic regulatory networks are considered, that model the post-transcriptional regulation of the PTEN onco-suppressor gene, mediated by microRNAs and competitive endogenous RNAs, in glioblastoma multiforme, the most severe of brain tumours. We simulate solutions of the resulting stochastic differential systems and discuss the effects of this miRNA-fashioned regulation on PTEN expression.Analysis of a new stochastic Gompertz diffusion model for untreated human glioblastomashttps://zbmath.org/1500.920232023-01-20T17:58:23.823708Z"Phan, Tuan Anh"https://zbmath.org/authors/?q=ai:phan.tuan-anh"Wang, Shuxun"https://zbmath.org/authors/?q=ai:wang.shuxun"Tian, Jianjun Paul"https://zbmath.org/authors/?q=ai:tian.jianjun-paulSummary: In this paper, we analyze a new Ito stochastic differential equation model for untreated human glioblastomas. The model was the best fit of the average growth and variance of 94 pairs of a data set. We show the existence and uniqueness of solutions in the positive spatial domain. When the model is restricted in the finite domain \((0,b)\), we show that the boundary point 0 is unattainable while the point \(b\) is reflecting attainable. We prove there is a unique ergodic stationary distribution for any non-zero noise intensity, and obtain the explicit probability density function for the stationary distribution. By using Brownian bridge, we give a representation of the probability density function of the first passage time when the diffusion process defined by a solution passes the point \(b\) firstly. We carry out numerical studies to illustrate our analysis. Our mathematical and numerical analysis confirms the soundness of our randomization of the deterministic model in that the stochastic model will set down to the deterministic model when the noise intensity approaches zero. We also give physical interpretation of our stochastic model and analysis.Using protein interactome similarity to improve \textit{random walk with restart model} for drug repurposinghttps://zbmath.org/1500.920272023-01-20T17:58:23.823708Z"Anjusha, I. T."https://zbmath.org/authors/?q=ai:anjusha.i-t"Saleena, N."https://zbmath.org/authors/?q=ai:saleena.n"Abdul Nazeer, K."https://zbmath.org/authors/?q=ai:abdul-nazeer.kSummary: Development of new drugs has the limitations of high cost, high time requirement and low success rate. If an existing drug can be used to treat a drug-seeking disease, we can reduce these limitations. The process of using existing drugs to treat new diseases is called drug repurposing. Since the drugs are already approved for human use, the success rate becomes high. In recent years, numerous random walk models have been proposed on the disease-drug heterogeneous network and become a popular drug repurposing approach. The performance of random walk-based approach depends on the network similarity measures used to build the heterogeneous network. In this paper, we improve the network similarity measures by integrating the similarity between the disease and drug-specific protein interactomes in human protein-protein interaction network. We then run a random walk with restart algorithm over the modified network to predict disease-drug relations. Our experiments reveal that performance of random walk model has improved after integrating protein interactome similarity.
For the entire collection see [Zbl 1491.65006].Lines of descent in the deterministic mutation-selection model with pairwise interactionhttps://zbmath.org/1500.920612023-01-20T17:58:23.823708Z"Baake, Ellen"https://zbmath.org/authors/?q=ai:baake.ellen"Cordero, Fernando"https://zbmath.org/authors/?q=ai:cordero.fernando"Hummel, Sebastian"https://zbmath.org/authors/?q=ai:hummel.sebastianSummary: We consider the mutation-selection differential equation with pairwise interaction (or, equivalently, the diploid mutation-selection equation) and establish the corresponding ancestral process, which is a random tree and a variant of the ancestral selection graph. The formal relation to the forward model is given via duality. To make the tree tractable, we prune branches upon mutations, thus reducing it to its informative parts. The hierarchies inherent in the tree are encoded systematically via tripod trees with weighted leaves; this leads to the stratified ancestral selection graph. The latter also satisfies a duality relation with the mutation-selection equation. Each of the dualities provides a stochastic representation of the solution of the differential equation. This allows us to connect the equilibria and their bifurcations to the long-term behaviour of the ancestral process. Furthermore, with the help of the stratified ancestral selection graph, we obtain explicit results about the ancestral type distribution in the case of unidirectional mutation.Exact computation of growth-rate variance in randomly fluctuating environmenthttps://zbmath.org/1500.920692023-01-20T17:58:23.823708Z"Unterberger, Jérémie"https://zbmath.org/authors/?q=ai:unterberger.jeremie-mSummary: We consider a general class of Markovian models describing the growth in a randomly fluctuating environment of a clonal biological population having several phenotypes related by stochastic switching. Phenotypes differ e.g. by the level of gene expression for a population of bacteria. The time-averaged growth rate of the population, \( \Lambda \), is self-averaging in the limit of infinite time; it may be understood as the fitness of the population in a context of Darwinian evolution. The observation time \(T\) being however typically finite, the growth rate fluctuates. For \(T\) finite but large, we obtain the variance of the time-averaged growth rate as the maximum of a functional based on the stationary probability distribution for the phenotypes. This formula is general. In the case of two states, the stationary probability was computed by
\textit{P. G. Hufton} et al. [J. Stat. Mech. Theory Exp. 2018, No. 2, Article ID 023501, 26 p. (2018; Zbl 1459.92061)], allowing for an explicit expression which can be checked numerically. Applications of our main formula to the study of survival strategies of biological populations, as developed in the companion article ([\textit{L. Dinis} et al., J. Stat. Mech. Theory Exp. 2022, No. 5, Article ID 053503, 24 p. (2022; Zbl 07565670)], are presented here briefly.Abstraction-guided truncations for stationary distributions of Markov population modelshttps://zbmath.org/1500.920852023-01-20T17:58:23.823708Z"Backenköhler, Michael"https://zbmath.org/authors/?q=ai:backenkohler.michael"Bortolussi, Luca"https://zbmath.org/authors/?q=ai:bortolussi.luca"Großmann, Gerrit"https://zbmath.org/authors/?q=ai:grossmann.gerrit"Wolf, Verena"https://zbmath.org/authors/?q=ai:wolf.verenaState-of-the-art methods for numerically calculating the stationary distribution of Markov population models rely on coarse truncations of irrelevant parts of large or infinite discrete state-spaces. These truncations are either obtained from the stationary statistical moments of the process or from Lyapunov theory. They are limited in shape because these methods do not take into account the detailed steady-state flow within the truncated state-space but only consider the average drift or stationary moments. In this article, the authors propose a method to find a tight truncation that is not limited in its shape and iteratively optimizes the set based on numerically cheap solutions of abstract intermediate models. It captures the main portion of probability mass even in the case of complex behaviors efficiently. In particular, the method represents another option, where Lyapunov analysis leads to forbiddingly large truncations. An aggregation scheme similar to the one used here has been previously proposed in [\textit{M. Backenköhler} et al., Lect. Notes Comput. Sci. 12651, 210--229 (2021; Zbl 1472.60140)] to analyze the bridging problem on Markov population models. This is the problem of analyzing process dynamics under both initial and terminal constraints.
For the entire collection see [Zbl 1482.68004].
Reviewer: Yingxin Guo (Qufu)Control in probability for SDE models of growth populationhttps://zbmath.org/1500.920932023-01-20T17:58:23.823708Z"Pérez-Aros, Pedro"https://zbmath.org/authors/?q=ai:perez-aros.pedro"Quiñinao, Cristóbal"https://zbmath.org/authors/?q=ai:quininao.cristobal"Tejo, Mauricio"https://zbmath.org/authors/?q=ai:tejo.mauricioSummary: In this paper, we consider a (control) optimization problem, which involves a stochastic dynamic. The model proposes selecting the best control function that keeps bounded a stochastic process over an interval of time with a high probability level. Here, the stochastic process is governed by a stochastic differential equation affected by a stochastic process. This setting becomes a chance-constrained control optimization problem, where the constraint is given by the probability level of infinitely many random inequalities. Since such a model is challenging, we discretize the dynamic and restrict the space of control functions to piecewise mappings. On the one hand, it transforms the infinite-dimensional optimization problem into a finite-dimensional one. On the other hand, it allows us to provide the well-posedness of the problem and approximation. Finally, the results are illustrated with numerical results, where classical model for the growth of a population are considered.Study of birth-death processes with immigrationhttps://zbmath.org/1500.920982023-01-20T17:58:23.823708Z"Viswanath, Narayanan"https://zbmath.org/authors/?q=ai:viswanath.narayanan-c"K. S., Shiny"https://zbmath.org/authors/?q=ai:k-s.shiny(no abstract)Exponential extinction of a stochastic predator-prey model with Allee effecthttps://zbmath.org/1500.920992023-01-20T17:58:23.823708Z"Zhang, Beibei"https://zbmath.org/authors/?q=ai:zhang.beibei"Wang, Hangying"https://zbmath.org/authors/?q=ai:wang.hangying"Lv, Guangying"https://zbmath.org/authors/?q=ai:lv.guangyingIn population dynamics, the Allee effect refers to a positive density dependence in prey population growth at small prey population sizes. Predator-prey model with Allee effect can describe more complex phenomenon. In this paper, the authors study a stochastic predator-prey model with Beddington-DeAngelis functional response and Allee effect, whose coefficients are dependent on expectations. The authors prove that there is a unique global positive solution to the system with the positive initial value. Moreover, sufficient conditions for exponential extinction are established. The main reference is the works of \textit{B. Tian} et al. [Int. J. Biomath. 8, No. 4, Article ID 1550044, 15 p. (2015; Zbl 1328.92070)].
Reviewer: Yingxin Guo (Qufu)Effect of movement on the early phase of an epidemichttps://zbmath.org/1500.921012023-01-20T17:58:23.823708Z"Arino, Julien"https://zbmath.org/authors/?q=ai:arino.julien"Milliken, Evan"https://zbmath.org/authors/?q=ai:milliken.evanSummary: The early phase of an epidemic is characterized by a small number of infected individuals, implying that stochastic effects drive the dynamics of the disease. Mathematically, we define the stochastic phase as the time during which the number of infected individuals remains small and positive. A continuous-time Markov chain model of a simple two-patch epidemic is presented. An algorithm for formalizing what is meant by \textit{small} is presented, and the effect of movement on the duration of the early stochastic phase of an epidemic is studied.Dynamics of a stochastic SIR epidemic model driven by Lévy jumps with saturated incidence rate and saturated treatment functionhttps://zbmath.org/1500.921052023-01-20T17:58:23.823708Z"EL Koufi, Amine"https://zbmath.org/authors/?q=ai:el-koufi.amine"Adnani, Jihad"https://zbmath.org/authors/?q=ai:adnani.jihad"Bennar, Abdelkrim"https://zbmath.org/authors/?q=ai:bennar.abdelkrim"Yousfi, Noura"https://zbmath.org/authors/?q=ai:yousfi.nouraSummary: In this article, we consider a stochastic SIR model with a saturated incidence rate and saturated treatment function incorporating Lévy noise. First, we prove the existence of a unique global positive solution to the model. We investigate the stability of the free equilibria \(E_0\) by using the Lyapunov method. We give sufficient conditions for the persistence in the mean. We show the dynamic properties of the solution around endemic equilibria point of the deterministic model. Moreover, we display some numerical results to confirm our theoretical results.A stochastic model of fowl pox disease: estimating the probability of disease outbreakhttps://zbmath.org/1500.921112023-01-20T17:58:23.823708Z"Muhumuza, Cosmas"https://zbmath.org/authors/?q=ai:muhumuza.cosmas"Mayambala, Fred"https://zbmath.org/authors/?q=ai:mayambala.fred"Mugisha, Joseph Y. T."https://zbmath.org/authors/?q=ai:mugisha.joseph-y-tThis paper presents a stochastic model of fowl pox disease focusing on estimating the probability of disease outbreak. A deterministic model is developed first, which describes the transmission dynamics of fowl pox disease. It is then transformed into a stochastic model using the continuous-time Markov chain approach. The total chicken population at time t is divided into three subpopulation classes, namely, the susceptible chicken \(S_c(t)\), infectious chicken \(I_c(t)\), and recovered chicken \(R_c(t)\). Under mosquito population, \(M(t)=S_m(t)+I_m(t)\), where \(M(t)\) is the total mosquito population at time \(t\), \(S_m(t)\) is the susceptible mosquito population and \(I_m(t)\) is the infected mosquito population. The system involves dynamics
\begin{align*}
dS_c/dt&=\lambda-\beta S_cI_c/N-\alpha S_cI_m/N-\sigma S_cB-\mu S_c,\\
dI_c/dt&=\beta S_cI_c/N+\alpha S_cI_m/N+\sigma S_cB-(\mu+\omega+\gamma)I_c,\\
dR_c/dt&=\gamma I_c-\mu R_c,\\
dS_m/dt&=\pi-\phi S_m I_c/N-\psi S_m,\\
dI_m/dt&=\phi S_mI_c/N-\psi I_m,\quad \text{ and } \\
dB/dt&=\tau I_c-\eta B,
\end{align*}
where \(N(t)=S_c(t)+I_c(t)+R_c(t)\) and \(B(t)\) is the concentration of fowl pox virus in the environment at time \(t\). The disease-free equilibrium, basic reproductive number, and endemic equilibrium point have been derived. Sensitivity analysis of the model parameters has also been studied.
Reviewer: Yilun Shang (Newcastle upon Tyne)Inference in Gaussian state-space models with mixed effects for multiple epidemic dynamicshttps://zbmath.org/1500.921122023-01-20T17:58:23.823708Z"Narci, Romain"https://zbmath.org/authors/?q=ai:narci.romain"Delattre, Maud"https://zbmath.org/authors/?q=ai:delattre.maud"Larédo, Catherine"https://zbmath.org/authors/?q=ai:laredo.catherine-f"Vergu, Elisabeta"https://zbmath.org/authors/?q=ai:vergu.elisabetaSummary: The estimation from available data of parameters governing epidemics is a major challenge. In addition to usual issues (data often incomplete and noisy), epidemics of the same nature may be observed in several places or over different periods. The resulting possible inter-epidemic variability is rarely explicitly considered. Here, we propose to tackle multiple epidemics through a unique model incorporating a stochastic representation for each epidemic and to jointly estimate its parameters from noisy and partial observations. By building on a previous work for prevalence data, a Gaussian state-space model is extended to a model with mixed effects on the parameters describing simultaneously several epidemics and their observation process. An appropriate inference method is developed, by coupling the SAEM algorithm with Kalman-type filtering. Moreover, we consider here incidence data, which requires to develop a new version of the filtering algorithm. Its performances are investigated on SIR simulated epidemics for prevalence and incidence data. Our method outperforms an inference method separately processing each dataset. An application to SEIR influenza outbreaks in France over several years using incidence data is also carried out. Parameter estimations highlight a non-negligible variability between influenza seasons, both in transmission and case reporting. The main contribution of our study is to rigorously and explicitly account for the inter-epidemic variability between multiple outbreaks, both from the viewpoint of modeling and inference with a parsimonious statistical model.Minimising the use of costly control measures in an epidemic elimination strategy: a simple mathematical modelhttps://zbmath.org/1500.921132023-01-20T17:58:23.823708Z"Plank, Michael J."https://zbmath.org/authors/?q=ai:plank.michael-jSummary: Countries such as New Zealand, Australia and Taiwan responded to the Covid-19 pandemic with an elimination strategy. This involves a combination of strict border controls with a rapid and effective response to eliminate border-related re-introductions. An important question for decision makers is, when there is a new re-introduction, what is the right threshold at which to implement strict control measures designed to reduce the effective reproduction number below 1. Since it is likely that there will be multiple re-introductions, responding at too low a threshold may mean repeatedly implementing controls unnecessarily for outbreaks that would self-eliminate even without control measures. On the other hand, waiting for too high a threshold to be reached creates a risk that controls will be needed for a longer period of time, or may completely fail to contain the outbreak. Here, we use a highly idealised branching process model of small border-related outbreaks to address this question. We identify important factors that affect the choice of threshold in order to minimise the expect time period for which control measures are in force. We find that the optimal threshold for introducing controls decreases with the effective reproduction number, and increases with overdispersion of the offspring distribution and with the effectiveness of control measures. Our results are not intended as a quantitative decision-making algorithm. However, they may help decision makers understand when a wait-and-see approach is likely to be preferable over an immediate response.A Markovian switching diffusion for an SIS model incorporating Lévy processeshttps://zbmath.org/1500.921162023-01-20T17:58:23.823708Z"Settati, A."https://zbmath.org/authors/?q=ai:settati.adel"Lahrouz, A."https://zbmath.org/authors/?q=ai:lahrouz.aadil"El Fatini, Mohamed"https://zbmath.org/authors/?q=ai:el-fatini.mohamed"El Haitami, A."https://zbmath.org/authors/?q=ai:el-haitami.a"El Jarroudi, M."https://zbmath.org/authors/?q=ai:el-jarroudi.moussa|el-jarroudi.mustapha"Erriani, M."https://zbmath.org/authors/?q=ai:erriani.mSummary: The purpose of this work is to investigate the asymptotic properties of a stochastic version of the classical SIS epidemic model with three noises. The stochastic model studied here includes white noise, telegraph noise and Lévy noise. We established conditions for extinction in probability and in \(pth\) moment. We also showed the persistence of disease under different conditions. The presented results are demonstrated by numerical simulations.Periodicity and stationary distribution of two novel stochastic epidemic models with infectivity in the latent period and household quarantinehttps://zbmath.org/1500.921172023-01-20T17:58:23.823708Z"Shangguan, Dongchen"https://zbmath.org/authors/?q=ai:shangguan.dongchen"Liu, Zhijun"https://zbmath.org/authors/?q=ai:liu.zhijun.1"Wang, Lianwen"https://zbmath.org/authors/?q=ai:wang.lianwen"Tan, Ronghua"https://zbmath.org/authors/?q=ai:tan.ronghua(no abstract)Carleman estimates of refined stochastic beam equations and applicationshttps://zbmath.org/1500.930122023-01-20T17:58:23.823708Z"Yu, Yongyi"https://zbmath.org/authors/?q=ai:yu.yongyi"Zhang, Ji-Feng"https://zbmath.org/authors/?q=ai:zhang.jifengSummary: This paper is devoted to establishing global Carleman estimates for refined stochastic beam equations. First, by establishing a fundamental weighted identity, two Carleman estimates are derived with different weight functions for the refined stochastic beam equation, which is a coupled system consisting of a stochastic ordinary differential equation and a stochastic partial differential equation. As applications of these Carleman estimates, the exact controllability of the refined system is proved by the least controls in some sense. Different from classical stochastic beam equations, the refined one is exactly controllable at any time. Meanwhile, the uniqueness of an inverse source problem for refined stochastic beam equations is obtained without any requirement on the initial and terminal data.Delay feedback stabilisation of stochastic differential equations driven by \(G\)-Brownian motionhttps://zbmath.org/1500.930932023-01-20T17:58:23.823708Z"Li, Yuyuan"https://zbmath.org/authors/?q=ai:li.yuyuan"Fei, Weiyin"https://zbmath.org/authors/?q=ai:fei.weiyin"Deng, Shounian"https://zbmath.org/authors/?q=ai:deng.shounianSummary: This paper aims to design the feedback control based on past states to stabilise a class of nonlinear stochastic differential equations driven by \(G\)-Brownian motion. By building up the connection between the delay feedback control and the control function of no-delay, sufficient conditions are established in terms of size of the time delay ensuring both the \(p\)th moment exponential stability and quasi-surely exponential stability of the delay feedback controlled system. Moreover, methods for determining the upper bound of the length of the time delay and further designing the delay feedback controller are provided. A numerical example is presented to demonstrate our new theory.A new result on stabilization analysis for stochastic nonlinear affine systems via Gamidov's inequalityhttps://zbmath.org/1500.931412023-01-20T17:58:23.823708Z"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Ezzine, Faten"https://zbmath.org/authors/?q=ai:ezzine.faten"Hammami, Mohamed Ali"https://zbmath.org/authors/?q=ai:hammami.mohamed-aliSummary: The Lyapunov approach is one of the most effective and efficient methods for the investigation of the stability of stochastic systems. Several authors analyzed the stability and stabilization of stochastic differential equations via Lyapunov techniques. Nevertheless, few results are concerned with the stability of stochastic systems based on the knowledge of the solution of the system explicitly. The originality of our work is to investigate the problem of stabilization of stochastic perturbed control-bilinear systems based on the explicit solution of the system by using the integral inequalities of the Gronwall type in particular Gamidov's inequality. Namely, under some restrictions on the perturbed term, and based on the method of integral inequalities, we prove that the stochastic system can be stabilized by constant feedback. Further, we study the problem of stabilization of stochastic perturbed control affine systems based on the use of bilinear approximation. Different examples are provided to verify the effectiveness of the proposed results.Maximum principle for optimal control of SPDEs with locally monotone coefficientshttps://zbmath.org/1500.931432023-01-20T17:58:23.823708Z"Coayla-Teran, Edson A."https://zbmath.org/authors/?q=ai:coayla-teran.edson-albertoSummary: The aim of this paper is to derive a maximum principle for a control problem governed by a stochastic partial differential equation (SPDE) with locally monotone coefficients. To reach our goal we adapt the method which uses the relation between backward stochastic partial differential equation (BSPDE) and the maximum principle. In particular, necessary conditions for optimality for this stochastic optimal control problem are obtained. In spite of the fact that the method used here was used by several authors before, our adaptation is not immediate. It applies a trick which is used to get estimates for solutions of SPDE with locally monotone coefficients as in the proof of the Lemmas 5.1 and 5.3. This adaptation permits us to apply our results to get a maximum principle for the optimal control to the cases when the system is governed by the 2D stochastic Navier-Stokes equation and by a stochastic Burgers' equation.Newton method for stochastic control problemshttps://zbmath.org/1500.931452023-01-20T17:58:23.823708Z"Gobet, Emmanuel"https://zbmath.org/authors/?q=ai:gobet.emmanuel"Grangereau, Maxime"https://zbmath.org/authors/?q=ai:grangereau.maximeSummary: We develop a new iterative method based on the Pontryagin principle to solve stochastic control problems. This method is nothing other than the Newton method extended to the framework of stochastic optimal control, where the state dynamics are given by an ODEs with stochastic coefficients and the cost is random. Each iteration of the method is made of two ingredients: computing the Newton direction, and finding an adapted step length. The Newton direction is obtained by solving an affine-linear forward-backward stochastic differential equation (FBSDE) with random coefficients. This is done in the setting of a general filtration. Solving such an FBSDE reduces to solving a Riccati backward stochastic differential equation (BSDE) and an affine-linear BSDE, as expected in the framework of linear FBSDEs or linear-quadratic stochastic control problems. We then establish convergence results for this Newton method. In particular, Lipschitz-continuity of the second-order derivative of the cost functional is established with an appropriate choice of norm and under boundedness assumptions, which is sufficient to prove (local) quadratic convergence of the method in the space of uniformly bounded processes. To choose an appropriate step length while fitting our choice of space of processes, an adapted backtracking line search method is developed. We then prove global convergence of the Newton method with the proposed line search procedure, which occurs at a quadratic rate after finitely many iterations. An implementation with regression techniques to solve BSDEs arising in the computation of the Newton step is developed. We apply it to the control problem of a large number of batteries providing ancillary services to an electricity network.A regularized stochastic subgradient projection method for an optimal control problem in a stochastic partial differential equationhttps://zbmath.org/1500.931462023-01-20T17:58:23.823708Z"Jadamba, Baasansuren"https://zbmath.org/authors/?q=ai:jadamba.baasansuren"Khan, Akhtar A."https://zbmath.org/authors/?q=ai:khan.akhtar-ali"Sama, Miguel"https://zbmath.org/authors/?q=ai:sama.miguelSummary: This work studies an optimal control problem in a stochastic partial differential equation. We present a new regularized stochastic subgradient projection iterative method for a general stochastic optimization problem. By using the martingale theory, we provide a convergence analysis for the proposed method. We test the iterative scheme's feasibility on the considered optimal control problem. The numerical results are encouraging and demonstrate the utility of a stochastic approximation framework in control problems with data uncertainty.
For the entire collection see [Zbl 1483.00042].Near optimality of stochastic control for singularly perturbed McKean-Vlasov systemshttps://zbmath.org/1500.931472023-01-20T17:58:23.823708Z"Li, Yun"https://zbmath.org/authors/?q=ai:li.yun"Wu, Fuke"https://zbmath.org/authors/?q=ai:wu.fuke"Zhang, Ji-feng"https://zbmath.org/authors/?q=ai:zhang.jifengSummary: In this paper, we are concerned with the optimal control problems for a class of systems with fast-slow processes. The problem under consideration is to minimize a functional subject to a system described by a two-time scaled McKean-Vlasov stochastic differential equation whose coefficients depend on state components and their probability distributions. Firstly, we establish the existence and uniqueness of the invariant probability measure for the fast process. Then, by using the relaxed control representation and the martingale method, we prove the weak convergence of the slow process and control process in the original problem, and we obtain an associated limit problem in which the coefficients are determined by the average of those of the original problem with respect to the invariant probability measure. Finally, by establishing the nearly optimal control of the limit problem, we obtain the near optimality of the original problem.Explicit solution to forward and backward stochastic differential equations with state delay and its application to optimal controlhttps://zbmath.org/1500.931482023-01-20T17:58:23.823708Z"Ma, Tianfu"https://zbmath.org/authors/?q=ai:ma.tianfu"Xu, Juanjuan"https://zbmath.org/authors/?q=ai:xu.juanjuan"Zhang, Huanshui"https://zbmath.org/authors/?q=ai:zhang.huanshui(no abstract)Analysis of artifacts in shell-based image inpainting: why they occur and how to eliminate themhttps://zbmath.org/1500.940032023-01-20T17:58:23.823708Z"Hocking, L. Robert"https://zbmath.org/authors/?q=ai:hocking.l-robert"Holding, Thomas"https://zbmath.org/authors/?q=ai:holding.thomas"Schönlieb, Carola-Bibiane"https://zbmath.org/authors/?q=ai:schonlieb.carola-bibianeSummary: In this paper we study a class of fast geometric image inpainting methods based on the idea of filling the inpainting domain in successive shells from its boundary inwards. Image pixels are filled by assigning them a color equal to a weighted average of their already filled neighbors. However, there is flexibility in terms of the order in which pixels are filled, the weights used for averaging, and the neighborhood that is averaged over. Varying these degrees of freedom leads to different algorithms, and indeed the literature contains several methods falling into this general class. All of them are very fast, but at the same time all of them leave undesirable artifacts such as ``kinking'' (bending) or blurring of extrapolated isophotes. Our objective in this paper is to build a theoretical model in order to understand why these artifacts occur and what, if anything, can be done about them. Our model is based on two distinct limits: a continuum limit in which the pixel width \(h \rightarrow 0\) and an asymptotic limit in which \(h > 0\) but \(h \ll 1\). The former will allow us to explain ``kinking'' artifacts (and what to do about them) while the latter will allow us to understand blur. Both limits are derived based on a connection between the class of algorithms under consideration and stopped random walks. At the same time, we consider a semi-implicit extension in which pixels in a given shell are solved for simultaneously by solving a linear system. We prove (within the continuum limit) that this extension is able to completely eliminate kinking artifacts, which we also prove must always be present in the direct method. Finally, we show that although our results are derived in the context of inpainting, they are in fact abstract results that apply more generally. As an example, we show how our theory can also be applied to a problem in numerical linear algebra.Robust recovery of sparse non-negative weights from mixtures of positive-semi-definite matriceshttps://zbmath.org/1500.940072023-01-20T17:58:23.823708Z"Jaensch, Fabian"https://zbmath.org/authors/?q=ai:jaensch.fabian"Jung, Peter"https://zbmath.org/authors/?q=ai:jung.peterSummary: We consider a structured estimation problem where an observed matrix is assumed to be generated as an \(s\)-sparse linear combination of \(N\) given \(n\times n\) positive-semi-definite matrices. Recovering the unknown \(N\)-dimensional and \(s\)-sparse weights from noisy observations is an important problem in various fields of signal processing and also a relevant preprocessing step in covariance estimation. We will present related recovery guarantees and focus on the case of non-negative weights. The problem is formulated as a convex program and can be solved without further tuning. Such robust, non-Bayesian and parameter-free approaches are important for applications where prior distributions and further model parameters are unknown. Motivated by explicit applications in wireless communication, we will consider the particular rank-one case, where the known matrices are outer products of iid. zero-mean sub-Gaussian \(n\)-dimensional complex vectors. We show that, for given \(n\) and \(N\), one can recover non-negative \(s\)-sparse weights with a parameter-free convex program once \(s\leq O(n^2/\log^2(N/n^2)\). Our error estimate scales linearly in the instantaneous noise power whereby the convex algorithm does not need prior bounds on the noise. Such estimates are important if the magnitude of the additive distortion depends on the unknown itself.