Recent zbMATH articles in MSC 60https://zbmath.org/atom/cc/602024-02-15T19:53:11.284213ZWerkzeugPartial shuffles by lazy swapshttps://zbmath.org/1526.050032024-02-15T19:53:11.284213Z"Janzer, Barnabás"https://zbmath.org/authors/?q=ai:janzer.barnabas"Johnson, J. Robert"https://zbmath.org/authors/?q=ai:johnson.j-robert"Leader, Imre"https://zbmath.org/authors/?q=ai:leader.imreSummary: How many random transpositions (meaning that we swap given pairs of elements with given probabilities independently) are needed to ensure that each element of \([n]\) is uniformly distributed -- in the sense that the probability that \(i\) is mapped to \(j\) is \(1/n\) for all \(i\) and \(j\)? And what if we insist that each pair is uniformly distributed? In this paper we show that the minimum for the first problem is about \(\frac{1}{2}n\log_2 n\), with this being exact when \(n\) is a power of 2. For the second problem, we show that, rather surprisingly, the answer is not quadratic: \(O(n\log^2n)\) random transpositions suffice. We also show that if we ask only that the pair \((1,2)\) is uniformly distributed, then the answer is \(2n-3\). This proves a conjecture of \textit{C. Groenland} et al. [``Short reachability networks'', Preprint, \url{arXiv:2208.06630}].Enumeration of three-quadrant walks via invariants: some diagonally symmetric modelshttps://zbmath.org/1526.050072024-02-15T19:53:11.284213Z"Bousquet-Mélou, Mireille"https://zbmath.org/authors/?q=ai:bousquet-melou.mireilleIn this paper, the author investigates a collection of eight models in the three-quadrant cone \(\mathcal{C} = \{(i, j): i\geq 0\) or \(j \geq 0\}\), which can be seen as the first level of difficulty beyond quadrant problems. The aim of this paper is to explore the applicability of invariants in the solution of three-quadrant models. This collection consists of diagonally symmetric models in \(\{-1, 0, 1\}^{2}\backslash \{(-1, 1), (1, -1)\}\). Three of them are known not to be D-algebraic. It is shown that the remaining five can be solved in a uniform fashion using Tutte's notion of invariants, which has already proved useful for some quadrant models. Three models are found to be algebraic, one is (only) D-finite, and the last one is (only) D-algebraic. The diagonal model \(\{\nearrow, \nwarrow, \swarrow, \searrow\}\) which is D-finite was solved in the same fashion. The three algebraic models are those of the Kreweras trilogy, \(\mathcal{S} = \{\nearrow, \leftarrow, \downarrow\}\), \(\mathcal{S}^{\ast} = \{\rightarrow, \uparrow, \swarrow\}\), and \(\mathcal{S}\cup \mathcal{S}^{\ast}\). The author's solution takes similar forms for all six models. Roughly speaking, the square of the generating function of three-quadrant walks with steps in \(\mathcal{S}\) is an explicit rational function in the quadrant generating function with steps in \(\{(j - i, j): (i, j)\in\mathcal{S}\}\). Various exact or asymptotic corollaries, including an explicit algebraic description of a positive harmonic function in \(\mathcal{C}\) for the (reverses of the) five models that are at least D-finite are deduced.
Reviewer: Ioan Tomescu (Bucureşti)On a matching arrangement of a graph and \(LP\)-orientations of a matching polyhedronhttps://zbmath.org/1526.050352024-02-15T19:53:11.284213Z"Bolotnikov, A. I."https://zbmath.org/authors/?q=ai:bolotnikov.a-i(no abstract)Normal approximation and fourth moment theorems for monochromatic triangleshttps://zbmath.org/1526.050452024-02-15T19:53:11.284213Z"Bhattacharya, Bhaswar B."https://zbmath.org/authors/?q=ai:bhattacharya.bhaswar-b"Fang, Xiao"https://zbmath.org/authors/?q=ai:fang.xiao|fang.xiao.1"Yan, Han"https://zbmath.org/authors/?q=ai:yan.hanSummary: Given a graph sequence \(\{ G_n\}_{n \geq 1}\) denote by \(T_3 (G_n)\) the number of monochromatic triangles in a uniformly random coloring of the vertices of \(G_n\) with \(c \geq 2\) colors. In this paper we prove a central limit theorem (CLT) for \(T_3 (G_n)\) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies \(c \geq 5\). We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any \(c \geq 2\), which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of \(T_3 (G_n)\), whenever \(c \geq 5\). Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [\textit{B. B. Bhattacharya} et al., Ann. Appl. Probab. 27, No. 1, 337--394 (2017; Zbl 1360.05051)].
{{\copyright} 2021 Wiley Periodicals LLC.}A fourth-moment phenomenon for asymptotic normality of monochromatic subgraphshttps://zbmath.org/1526.050462024-02-15T19:53:11.284213Z"Das, Sayan"https://zbmath.org/authors/?q=ai:das.sayan-kumar"Himwich, Zoe"https://zbmath.org/authors/?q=ai:himwich.zoe-m"Mani, Nitya"https://zbmath.org/authors/?q=ai:mani.nityaSummary: Given a graph sequence \(\left\{ G_n\right\}_{n\geq 1}\) and a simple connected subgraph \(H\), we denote by \(T\left(H,G_n\right)\) the number of monochromatic copies of \(H\) in a uniformly random vertex coloring of \(G_n\) with \(c\geq 2\) colors. We prove a central limit theorem for \(T\left(H,G_n\right)\) (we denote the appropriately centered and rescaled statistic as \(Z\left(H,G_n\right))\) with explicit error rates. The error rates arise from graph counts of collections formed by joining copies of \(H\) which we call good joins. Good joins are closely related to the fourth moment of \(Z\left(H,G_n\right)\), which allows us to show a fourth moment phenomenon for the central limit theorem. For \(c\geq 30\), we show that \(Z\left(H,G_n\right)\) converges in distribution to \(\mathcal{N} (0,1)\) whenever its fourth moment converges to 3. We show the convergence of the fourth moment is necessary to obtain a normal limit when \(c\geq 2\).
{\copyright} 2023 Wiley Periodicals LLC.The rank of the sandpile group of random directed bipartite graphshttps://zbmath.org/1526.050632024-02-15T19:53:11.284213Z"Bhargava, Atal"https://zbmath.org/authors/?q=ai:bhargava.atal"DePascale, Jack"https://zbmath.org/authors/?q=ai:depascale.jack"Koenig, Jake"https://zbmath.org/authors/?q=ai:koenig.jakeSummary: We identify the asymptotic distribution of \(p\)-rank of the sandpile group of random directed bipartite graphs which are not too imbalanced. We show this matches exactly with that of the Erdös-Rényi random directed graph model, suggesting that the Sylow \(p\)-subgroups of this model may also be Cohen-Lenstra distributed. Our work builds on the results of \textit{S. Koplewitz} [Ann. Comb. 27, No. 1, 1--18 (2023; Zbl 1518.05171)] who studied \(p\)-rank distributions for unbalanced random bipartite graphs, and showed that for sufficiently unbalanced graphs, the distribution of \(p\)-rank differs from the Cohen-Lenstra distribution. Koplewitz [loc. cit.] conjectured that for random balanced bipartite graphs, the expected value of \(p\)-rank is \(O(1)\) for any \(p\). This work proves his conjecture and gives the exact distribution for the subclass of directed graphs.Non-linear log-Sobolev inequalities for the Potts semigroup and applications to reconstruction problemshttps://zbmath.org/1526.051022024-02-15T19:53:11.284213Z"Gu, Yuzhou"https://zbmath.org/authors/?q=ai:gu.yuzhou"Polyanskiy, Yury"https://zbmath.org/authors/?q=ai:polyanskiy.yurySummary: Consider the semigroup of random walk on a complete graph, which we call the Potts semigroup. \textit{P. Diaconis} and \textit{L. Saloff-Coste} [Ann. Appl. Probab. 6, No. 3, 695--750 (1996; Zbl 0867.60043)] computed the maximum ratio between the relative entropy and the Dirichlet form, obtaining the constant \(\alpha_2\) in the 2-log-Sobolev inequality (2-LSI). In this paper, we obtain the best possible non-linear inequality relating entropy and the Dirichlet form (i.e., \(p\)-NLSI, \(p \geq 1\)). As an example, we show \(\alpha_1 = 1+\frac{1+o(1)}{\log q}\). Furthermore, \(p\)-NLSIs allow us to conclude that for \(q \geq 3\), distributions that are not a product of identical distributions can have slower speed of convergence to equilibrium, unlike the case \(q=2\). By integrating the 1-NLSI we obtain new strong data processing inequalities (SDPI), which in turn allows us to improve results of \textit{E. Mossel} and \textit{Y. Peres} [ibid. 13, No. 3, 817--844 (2003; Zbl 1050.60082)] on reconstruction thresholds for Potts models on trees. A special case is the problem of reconstructing color of the root of a \(q\)-colored tree given knowledge of colors of all the leaves. We show that to have a non-trivial reconstruction probability the branching number of the tree should be at least
\[
\frac{\log q}{\log q - \log (q-1)} = (1-o(1))q \log q.
\]
This recovers previous results (of \textit{A. Sly} [Commun. Math. Phys. 288, No. 3, 943--961 (2009; Zbl 1274.60163)] and \textit{N. Bhatnagar} et al. [SIAM J. Discrete Math. 25, No. 2, 809--826 (2011; Zbl 1298.05060)]) in (slightly) more generality, but more importantly avoids the need for any coloring-specific arguments. Similarly, we improve the state-of-the-art on the weak recovery threshold for the stochastic block model with \(q\) balanced groups, for all \(q \geq 3\). To further show the power of our method, we prove optimal non-reconstruction results for a broadcasting on trees model with Gaussian kernels, closing a gap left open by \textit{R. Eldan} et al. [Comb. Probab. Comput. 31, No. 6, 1048--1069 (2022; Zbl 07671090)]. These improvements advocate information-theoretic methods as a useful complement to the conventional techniques originating from the statistical physics.Concentration estimates for functions of finite high-dimensional random arrayshttps://zbmath.org/1526.051242024-02-15T19:53:11.284213Z"Dodos, Pandelis"https://zbmath.org/authors/?q=ai:dodos.pandelis"Tyros, Konstantinos"https://zbmath.org/authors/?q=ai:tyros.konstantinos"Valettas, Petros"https://zbmath.org/authors/?q=ai:valettas.petrosSummary: Let \(\mathfrak{X}\) be a \(d\)-dimensional random array on \([n]\) whose entries take values in a finite set \(\mathcal{X}\), that is, \(\mathfrak{X}=\langle X_s :s\in {\binom{[n]}{d}}\rangle\) is an
\(\mathcal{X}\)-valued stochastic process indexed by the set \(\binom{[n]}{d}\) of all \(d\)-element subsets of \([n]:= \{1,\ldots, n\}\). We give easily checked conditions on \(\mathfrak{X}\) that ensure, for instance, that for every function \(f : \mathcal{X}^{\binom{[n]}{d}} \to \mathbb{R}\) that satisfies \(\mathbb{E} [f(\mathfrak{X})] = 0\) and \(\Vert f(\mathfrak{X})\Vert_{L_p}=1\) for some \(p>1\), the random variable \(f(\mathfrak{X})\) becomes concentrated after conditioning it on a large subarray of \(\mathfrak{X}\). These conditions cover several classes of random arrays with not necessarily independent entries. Applications are given in combinatorics, and examples are also presented that show the optimality of various aspects of the results.
{\copyright} 2023 The Authors. \textit{Random Structures \& Algorithms} published by Wiley Periodicals LLC.Color-avoiding percolation of random graphs: between the subcritical and the intermediate regimehttps://zbmath.org/1526.051252024-02-15T19:53:11.284213Z"Lichev, Lyuben"https://zbmath.org/authors/?q=ai:lichev.lyubenThe paper under review is concerned with colour-avoiding (CA) percolation. Here, every edge of a graph \(G=(V,E)\) receives some of \(k\) colours. We say two vertices \(u\) and \(v\) of \(G\) are CA-connected if they can be connected using every \(k-1\) element subset of the \(k\) colours. This is clearly an equivalence relation on \(V\) and the equivalence classes are called CA components. We denote the CA component of a vertex \(u\) by \(\tilde{C}(u)\).
In [\textit{L. Lichev} and \textit{B. Schapira}, ``Color-avoiding percolation on the Erdős-Rényi random graph'', Preprint, \url{arXiv:2211.16086}], this is studied for a randomly coloured Erdős-Rényi random graph of constant average degree. It is shown that there are three regimes for this: a supercritical regime in which the CA components have order linear in the number of vertices of \(G\), an intermediate regime in which the CA components have order logarithmic in the number of vertices, and a subcritical regime in which the CA components have a size which is bounded, indeed bounded by the number of colours.
Formally, let \(\lambda_{1}\geq \lambda_{2}\dots \geq \lambda_{k}\) be fixed, and \(G_{i}\) be a random graph \(G(n,\lambda_{i}/n)\) (all of these on vertex set \([n]\)) thought of as the edges of colour \(i\). Let \(G\) be the union of these graphs and \(G^{\ell}\) the union of the \(G_{i}\) for all \(1\leq i\leq k\) with \(i\neq \ell\). Similarly let \(\Lambda=\sum_{i=1}^{k}\lambda_{i}\) and \(\lambda_{i}^{\ast}=\Lambda-\lambda_{i}\) for every \(i\) so that the \(\lambda_{i}^{\ast}\) are now an increasing sequence. Then the result is that \(\max_{u\in [n]}\vert \tilde{C}(u)\vert/n\) converges in probability to a constant \(a_{1}\), which is strictly positive if and only if \(\lambda_{1}^{\ast}>1\). If instead we have \(\lambda_{k}^{\ast}>1>\lambda_{k-1}^{\ast}\), then \(\max_{u\in V}\vert \tilde{C}(u)\vert/\log(n)\) converges in probability to a constant \(a_{2}\) and finally if \(\lambda_{k}^{\ast}<1\) then the largest CA component order is bounded above by \(k\) with probability tending to 1.
The main contribution of the paper under review is to understand more fully the transition between the subcritical and intermediate regimes. In detail, let \(\zeta=\zeta(n)\) tend to 0 as \(n\rightarrow\infty\). Let \(\lambda_{k}^{\ast}=1+\zeta\) and \(\lambda_{k-1}^{\ast}<1\). Then if \(\log(\zeta^{-1})/\log(n)\rightarrow 0\) as \(n\rightarrow\infty\) we have that \(\max_{u\in [n]}\vert \tilde{C}(u)\vert \log(\zeta^{-1})/\log(n)\) converges in probability to 1. However if instead there is \(\varepsilon>0\) such that \(\zeta\leq n^{-\varepsilon}\) for large enough \(n\), then \(\sup_{n}\mathbb{P}(\max_{u\in [m]}\vert \tilde{C}(u)\vert \geq M)\) tends to 0 as \(M\rightarrow \infty\).
Proofs use detailed information on the component sizes in Erdős-Rényi random graphs, but there is also a non-trivial optimisation step involved.
Reviewer: David B. Penman (Colchester)Abelian groups from random hypergraphshttps://zbmath.org/1526.051262024-02-15T19:53:11.284213Z"Newman, Andrew"https://zbmath.org/authors/?q=ai:newman.andrew-fSummary: For a \(k\)-uniform hypergraph \(\mathcal{H}\) on vertex set \(\{1, \ldots, n\}\) we associate a particular signed incidence matrix \(M(\mathcal{H})\) over the integers. For \(\mathcal{H} \sim \mathcal{H}_k(n, p)\) an Erdős-Rényi random \(k\)-uniform hypergraph, \({\mathrm{coker}}(M(\mathcal{H}))\) is then a model for random abelian groups. Motivated by conjectures from the study of random simplicial complexes we show that for \(p = \omega (1/n^{k - 1}), {\mathrm{coker}}(M(\mathcal{H}))\) is torsion-free.Multiple random walks on graphs: mixing few to cover manyhttps://zbmath.org/1526.051282024-02-15T19:53:11.284213Z"Rivera, Nicolás"https://zbmath.org/authors/?q=ai:rivera.nicolas"Sauerwald, Thomas"https://zbmath.org/authors/?q=ai:sauerwald.thomas"Sylvester, John"https://zbmath.org/authors/?q=ai:sylvester.johnSummary: Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running \(k\) multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by \textit{N. Alon} et al. [ibid. 20, No. 4, 481--502 (2011; Zbl 1223.05284)]) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when \(k\) random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of \(\Omega ((n/k) \log n)\) on the stationary cover time, holding for any \(n\)-vertex graph \(G\) and any \(1 \leq k =o(n\log n)\). Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.Reversible random walks on dynamic graphshttps://zbmath.org/1526.051292024-02-15T19:53:11.284213Z"Shimizu, Nobutaka"https://zbmath.org/authors/?q=ai:shimizu.nobutaka"Shiraga, Takeharu"https://zbmath.org/authors/?q=ai:shiraga.takeharuSummary: This paper discusses random walks on edge-changing dynamic graphs. We prove general and improved bounds for mixing, hitting, and cover times for a random walk according to a sequence of irreducible and reversible transition matrices with the time-invariant stationary distribution. An interesting consequence is the tight bounds of the lazy Metropolis walk on any dynamic connected graph. We also prove bounds for multiple random walks on dynamic graphs. Our results extend previous upper bounds for simple random walks on dynamic graphs and give improved and tight upper bounds in several cases. Our results reinforce the observation that time-inhomogeneous Markov chains with an invariant stationary distribution behave almost identically to a time-homogeneous chain.
{\copyright} 2023 Wiley Periodicals LLCHydrodynamic limit of the Robinson-Schensted-Knuth algorithmhttps://zbmath.org/1526.051382024-02-15T19:53:11.284213Z"Marciniak, Mikołaj"https://zbmath.org/authors/?q=ai:marciniak.mikolajSummary: We investigate the evolution in time of the position of a fixed number in the insertion tableau when the Robinson-Schensted-Knuth algorithm is applied to a sequence of random numbers. When the length of the sequence tends to infinity, a typical trajectory after scaling converges uniformly in probability to some deterministic curve.
{{\copyright} 2021 Wiley Periodicals LLC.}Law of large numbers for Betti numbers of homogeneous and spatially independent random simplicial complexeshttps://zbmath.org/1526.051402024-02-15T19:53:11.284213Z"Kanazawa, Shu"https://zbmath.org/authors/?q=ai:kanazawa.shuSummary: The Linial-Meshulam complex model is a natural higher dimensional analog of the Erdős-Rényi graph model. In recent years, \textit{N. Linial} and \textit{Y. Peled} [Ann. Math. (2) 184, No. 3, 745--773 (2016; Zbl 1348.05193); in: A journey through discrete mathematics. A tribute to Jiří Matoušek. Cham: Springer. 543--570 (2017; Zbl 1380.05205)] established a limit theorem for Betti numbers of Linial-Meshulam complexes [\textit{N. Linial} and \textit{R. Meshulam}, Combinatorica 26, No. 4, 475--487 (2006; Zbl 1121.55013)] with an appropriate scaling of the underlying parameter. The present article aims to extend that result to more general random simplicial complex models. We introduce a class of homogeneous and spatially independent random simplicial complexes, including the Linial-Meshulam complex model and the random clique complex model as special cases, and we study the asymptotic behavior of their Betti numbers. Moreover, we obtain the convergence of the empirical spectral distributions of their Laplacians. A key element in the argument is the local weak convergence of simplicial complexes. Inspired by the work of Linial and Peled [loc. cit.], we establish the local weak limit theorem for homogeneous and spatially independent random simplicial complexes.
{{\copyright} 2021 Wiley Periodicals LLC.}Hilbert evolution algebras and its connection with discrete-time Markov chainshttps://zbmath.org/1526.170512024-02-15T19:53:11.284213Z"Vidal, Sebastian J."https://zbmath.org/authors/?q=ai:vidal.sebastian-j"Cadavid, Paula"https://zbmath.org/authors/?q=ai:cadavid.paula"Rodriguez, Pablo M."https://zbmath.org/authors/?q=ai:rodriguez.pablo-mIn this paper, the authors introduce Hilbert evolution algebras as an extension of evolution algebras in a separable Hilbert space, so that an infinite number of non-zero structure constants is feasible. Then, they define the evolution operator of these new algebras, and study under which conditions it is continuous. The iterative application of this operator provides certain boundedness conditions on transition probabilities so that the whole dynamics of a discrete-time Markov chain with infinite countable state space can be described by a particular type of Hilbert evolution algebra. Some properties concerning the boundedness of the evolution operator of this algebra are studied.
Reviewer: Raúl M. Falcón (Sevilla)Dimensions of fractional Brownian imageshttps://zbmath.org/1526.280032024-02-15T19:53:11.284213Z"Burrell, Stuart A."https://zbmath.org/authors/?q=ai:burrell.stuart-aSummary: This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-\( \alpha\) fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.A probabilistic proof of an asymptotic formula for the modified Bessel functionhttps://zbmath.org/1526.330012024-02-15T19:53:11.284213Z"Fejzullahu, Bujar Xh."https://zbmath.org/authors/?q=ai:fejzullahu.bujar-xhSummary: In this paper the local limit theorem for lattice distributions has been applied to deduce the asymptotic behavior (as \(n\rightarrow \infty)\) for the modified Bessel function \(I_{\nu -\beta_n}(2x\sqrt{\lambda \alpha_n})\), where \((\alpha_n)_{n\in\mathbb{N}}\) and \((\beta_n)_{n\in\mathbb{N}}\) are any two increasing sequences of natural numbers such that \(\frac{(\lambda \alpha_n -\beta_n)^2}{2\lambda \alpha_n}\rightarrow \mu_{\lambda} \geq 0\) as \(n\rightarrow \infty, \lambda >0\), and \(\nu \geq 0\). Our asymptotics is uniformly valid in the compact subsets of \((0,\infty)\).Multiple hypergeometric functions in communication theory: Evaluations of error probabilities for four-parameter, \( \kappa{-}\mu\) and \(\eta{-}\mu\) signals distributions in general fading channelshttps://zbmath.org/1526.330022024-02-15T19:53:11.284213Z"Savischenko, N. V."https://zbmath.org/authors/?q=ai:savischenko.nikolay-v"Brychkov, Yu. A."https://zbmath.org/authors/?q=ai:brychkov.yury-aSummary: Mathematical formulation of the problem of calculating the probability of erroneous reception of multi-position signals in a communication channel with general fading, described by four-parameter (within the framework of the Gaussian model of the communication channel) \({\kappa{-}\mu}, \eta{-}\mu\) distributions and additive white Gaussian noise (AWGN), is presented. The relationship between solution of the problem and the theory of multiple hypergeometric functions is shown. The four-parameter distribution is considered which includes three-parameter distribution (Beckmann distribution); two-parameter Hoyt distribution (Nakagami-\(q)\); Rice distribution (Nakagami-\(n)\); Rayleigh distribution; one-sided normal distribution with zero variance and zero expectation. Formulas for the initial moments of the four-parameter distribution are given in integral and series form. A method for calculating symbol and bit error probabilities in communication system with multi-position signals is proposed. It is shown that for the four-parameter, \({\kappa{-}\mu}\) and \(\eta{-}\mu\) distributions, the calculation of the error probability can be reduced to the application of a new special \(\mathcal{S} \)-function. For a family of generalized \({\kappa{-}\mu}\) distributions, formulas for the probability distribution density containing multiple hypergeometric functions are given. Formulas are obtained for the initial moments of the generalized \({\kappa{-}\mu}\) distribution.Critical edge behavior in the singularly perturbed Pollaczek-Jacobi type unitary ensemblehttps://zbmath.org/1526.330032024-02-15T19:53:11.284213Z"Wang, Zhaoyu"https://zbmath.org/authors/?q=ai:wang.zhaoyu"Fan, Engui"https://zbmath.org/authors/?q=ai:fan.enguiSummary: We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek-Jacobi type weight
\[
w_{\mathrm{PJ} 2}(x,t;\alpha,\beta)= x^\alpha (1-x)^\beta e^{- \frac{t}{x (1-x)}},
\]
where \(t\in[0,\infty)\), \(\alpha>0\), \(\beta>0\) and \(0<x<1\). Based on our observation, we find that this weight includes the symmetric constraint \(w_{P J 2}(x,t;\alpha,\beta)= w_{P J 2}(1-x,t;\beta,\alpha)\). Our main results obtained here include two aspects: (1) Strong asymptotics: we deduce strong asymptotics of monic orthogonal polynomials with respect to the above weight function in different regions in the complex plane when the polynomial degree \(n\) goes to infinity. Because of the effect of \(\frac{t}{x (1-x)}\) for varying \(t\), the asymptotic behavior in a neighborhood of point \(1\) is described in terms of the Airy function as \(\zeta=2 n^2t\to\infty\), \(n\to\infty \), but the Bessel function as \(\zeta\to 0\), \(n\to\infty \). Due to symmetry, the similar local asymptotic behavior near the singular point \(x=0\) can be derived. (2) Limiting eigenvalue correlation kernels: We calculate the limit of the eigenvalue correlation kernel of the corresponding unitary random matrix ensemble in the bulk of the spectrum described by the sine kernel, and at both sides of hard edge, expressed as a Painlevé III kernel. Our analysis is based on the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems.A converse to the neo-classical inequality with an application to the Mittag-Leffler functionhttps://zbmath.org/1526.330042024-02-15T19:53:11.284213Z"Gerhold, Stefan"https://zbmath.org/authors/?q=ai:gerhold.stefan"Simon, Thomas"https://zbmath.org/authors/?q=ai:simon.thomasSummary: We prove two inequalities for the Mittag-Leffler function, namely that the function \(\log E_\alpha (x^\alpha)\) is sub-additive for \(0 < \alpha < 1\), and super-additive for \(\alpha >1\). These assertions follow from two new binomial inequalities, one of which is a converse to the neo-classical inequality. The proofs use a generalization of the binomial theorem due to Hara and Hino (Bull London Math Soc 2010). For \(0 < \alpha < 2\), we also show that \(E_\alpha (x^\alpha)\) is log-concave resp. log-convex, using analytic as well as probabilistic arguments.Approximate controllability for impulsive stochastic delayed differential inclusionshttps://zbmath.org/1526.340562024-02-15T19:53:11.284213Z"Yadav, Shobha"https://zbmath.org/authors/?q=ai:yadav.shobha"Kumar, Surendra"https://zbmath.org/authors/?q=ai:kumar.surendra-shashiSummary: This manuscript is devoted to the approximate controllability of semilinear infinite-delayed stochastic differential inclusions involving impulses. The presence of a mild solution for the aforementioned system is shown by employing the theory of stochastic analysis and the fixed point theorem due to Covitz and Nadler. Then, we construct a group of sufficient conditions and utilize Grownwall's inequality to establish the approximate controllability results. Finally, two examples are constructed to exemplify the feasibility of derived outcomes.Ergodic stationary distribution and extinction of stochastic delay chemostat system with Monod-Haldane functional response and higher-order Lévy jumpshttps://zbmath.org/1526.340612024-02-15T19:53:11.284213Z"Chen, Xingzhi"https://zbmath.org/authors/?q=ai:chen.xingzhi"Li, Dong"https://zbmath.org/authors/?q=ai:li.dong.2|li.dong.3|li.dong|li.dong.5|li.dong.1"Tian, Baodan"https://zbmath.org/authors/?q=ai:tian.baodan"Yang, Dan"https://zbmath.org/authors/?q=ai:yang.danSummary: In this work, the case of the stochastic delayed chemostat model with Monod-Haldane functional response and nonlinear stochastic perturbation is investigated, in which both higher-order Gaussian white noise and Lévy jumps are introduced. Firstly, the condition for the occurrence of Hopf bifurcation in the deterministic system is derived. Secondly, the existence and uniqueness of the positive global solution of the stochastic system are established, and the stochastically ultimate boundedness of the solution is studied. To understand the statistical characteristics of the stochastic system, the key threshold is established by investigating the auxiliary equation of the corresponding stochastic system. Then, the existence of an ergodic stationary distribution of the stochastic system, which is indicative of the long-term persistence of the microbial population from a biological perspective, is proved. Furthermore, the extinction of the microorganism \(x\) for the stochastic system is studied. Finally, some important numerical experiments are provided to further support the theoretical results. The impact of various noises and time delays on the stochastic system is also analyzed using 2D and 3D graphs of the joint two-dimensional densities.Large-time behaviour for anisotropic stable nonlocal diffusion problems with convectionhttps://zbmath.org/1526.350582024-02-15T19:53:11.284213Z"Endal, Jørgen"https://zbmath.org/authors/?q=ai:endal.jorgen"Ignat, Liviu I."https://zbmath.org/authors/?q=ai:ignat.liviu-i"Quirós, Fernando"https://zbmath.org/authors/?q=ai:quiros-gracian.fernandoSummary: We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable Lévy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution, with the same mass, of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a ``projection'' of the original one onto the subspace. The determination of this projection is one of the main issues of the paper. When convection and diffusion are of the same order the limit equation coincides with the original one.
Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We are able to cover both the cases of slow and fast convection, as long as the mass is preserved. Fast convection, which corresponds to convection nonlinearities that are not locally Lipschitz, but only locally Hölder, has not been considered before in the nonlocal diffusion setting.Stability of traveling waves for deterministic and stochastic delayed reaction-diffusion equation based on phase shifthttps://zbmath.org/1526.351102024-02-15T19:53:11.284213Z"Liu, Yu"https://zbmath.org/authors/?q=ai:liu.yu.16"Chen, Guanggan"https://zbmath.org/authors/?q=ai:chen.guanggan"Li, Shuyong"https://zbmath.org/authors/?q=ai:li.shuyongSummary: In this paper, we establish the nonlinear orbital stability of the traveling wave solution of deterministic and stochastic delayed reaction-diffusion equation. Employing the deterministic phase shift and establishing a delayed-integral inequality, we obtain the exponential stability of the traveling wave solution for the deterministic delayed reaction-diffusion equation. Applying a stochastic phase shift and time transformation, we then verify that the traveling wave solution of the deterministic equation retain the nonlinear orbital stability when the noise intensity is small enough and the initial value sufficiently closes the traveling wave.Fractional Kolmogorov operator and desingularizing weightshttps://zbmath.org/1526.352032024-02-15T19:53:11.284213Z"Kinzebulatov, Damir"https://zbmath.org/authors/?q=ai:kinzebulatov.damir"Semënov, Yuliy A."https://zbmath.org/authors/?q=ai:semenov.yuliy-aSummary: We establish sharp upper and lower bounds on the heat kernel of the fractional Laplace operator perturbed by Hardy-type drift by transferring it to an appropriate weighted space with singular weight.Global null-controllability for stochastic semilinear parabolic equationshttps://zbmath.org/1526.352152024-02-15T19:53:11.284213Z"Hernández-Santamaría, Víctor"https://zbmath.org/authors/?q=ai:hernandez-santamaria.victor"Le Balc'h, Kévin"https://zbmath.org/authors/?q=ai:le-balch.kevin"Peralta, Liliana"https://zbmath.org/authors/?q=ai:peralta.lilianaAuthors' abstract: In this paper we prove the small-time global null-controllability of forward (respectively backward) semilinear stochastic parabolic equations with globally Lipschitz nonlinearities in the drift and the diffusion terms (respectively in the drift term). In particular, we solve the open question posed by \textit{S. Tang} and \textit{X. Zhang} [SIAM J. Control Optim. 48, No. 4, 2191--2216 (2009; Zbl 1203.93027)]. We propose a new twist on a classical strategy for controlling linear stochastic systems. By employing a new refined Carleman estimate, we obtain a controllability result in a weighted space for a linear system with source terms. The main novelty here is that the Carleman parameters are made explicit and are then used in a Banach fixed point method. This allows us to circumvent the well-known problem of the lack of compactness embeddings for the solutions spaces arising in the study of controllability problems for stochastic PDEs.
Reviewer: Kaïs Ammari (Monastir)A short remark on inviscid limit of the stochastic Navier-Stokes equationshttps://zbmath.org/1526.352582024-02-15T19:53:11.284213Z"Chaudhary, Abhishek"https://zbmath.org/authors/?q=ai:chaudhary.abhishek"Vallet, Guy"https://zbmath.org/authors/?q=ai:vallet.guySummary: In this article, we study the inviscid limit of the stochastic incompressible Navier-Stokes equations in three-dimensional space. We prove that a subsequence of weak martingale solutions of the stochastic incompressible Navier-Stokes equations converges strongly to a weak martingale solution of the stochastic incompressible Euler equations in the periodic domain under the well-accepted hypothesis, namely Kolmogorov hypothesis [\textit{A. N. Kolmogorov}, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 32, 16--18 (1941; Zbl 0063.03292); ibid. 31, 538--540 (1941; Zbl 0026.17001); ibid. 30, 301--305 (1941; Zbl 0025.37602; JFM 67.0850.06)].Rigorous derivation of the degenerate parabolic-elliptic Keller-Segel system from a moderately interacting stochastic particle system. I: Partial differential equationhttps://zbmath.org/1526.352742024-02-15T19:53:11.284213Z"Chen, Li"https://zbmath.org/authors/?q=ai:chen.li.1"Gvozdik, Veniamin"https://zbmath.org/authors/?q=ai:gvozdik.veniamin"Li, Yue"https://zbmath.org/authors/?q=ai:li.yueIn the first part of the study on the approximation of the parabolic-elliptic Keller-Segel system with degenerate diffusion by stochastic interacting particles, the authors discuss a nonlocal regularized parabolic system. Among the parabolic pde theory tools Bernstein estimates play an important role in the derivation of properties of weak solutions. These results are used in the next part of the work to pass to the mean field limit in the stochastic particles system.
Reviewer: Piotr Biler (Wrocław)Recurrence of the random process governed with the fractional Laplacian and the Caputo time derivativehttps://zbmath.org/1526.352812024-02-15T19:53:11.284213Z"Affili, Elisa"https://zbmath.org/authors/?q=ai:affili.elisa"Kemppainen, Jukka T."https://zbmath.org/authors/?q=ai:kemppainen.jukka-tSummary: We are addressing a parabolic equation with fractional derivatives in time and space that governs the scaling limit of continuous-time random walks with anomalous diffusion. For these equations, the fundamental solution represents the probability density of finding a particle released at the origin at time 0 at a given position and time. Using some estimates of the asymptotic behaviour of the fundamental solution, we evaluate the probability of the process returning infinite times to the origin in a heuristic way. Our calculations suggest that the process is always recurrent.
For the entire collection see [Zbl 1522.35003].On initial value problem for elliptic equation on the plane under Caputo derivativehttps://zbmath.org/1526.352862024-02-15T19:53:11.284213Z"Binh, Tran Thanh"https://zbmath.org/authors/?q=ai:binh.tran-thanh"Thang, Bui Dinh"https://zbmath.org/authors/?q=ai:thang.bui-dinh"Phuong, Nguyen Duc"https://zbmath.org/authors/?q=ai:phuong.nguyen-ducSummary: In this article, we are interested to study the elliptic equation under the Caputo derivative. We obtain several regularity results for the mild solution based on various assumptions of the input data. In addition, we derive the lower bound of the mild solution in the appropriate space. The main tool of the analysis estimation for the mild solution is based on the bound of the Mittag-Leffler functions, combined with analysis in Hilbert scales space. Moreover, we provide a regularized solution for our problem using the Fourier truncation method. We also obtain the error estimate between the regularized solution and the mild solution. Our current article seems to be the first study to deal with elliptic equations with Caputo derivatives on the unbounded domain.Stochastic resetting and linear reaction processes: a continuous time random walk approachhttps://zbmath.org/1526.353432024-02-15T19:53:11.284213Z"da Rocha, Gabriel G."https://zbmath.org/authors/?q=ai:da-rocha.gabriel-g"Lenzi, Ervin K."https://zbmath.org/authors/?q=ai:kaminski-lenzi.ervinSummary: We investigate a diffusion process by considering simultaneously stochastic resetting and linear reaction kinetics in the continuous time random walk approach. We first consider the formalism for a single species and then extend it to multiple species. We perform the analysis by considering a general probability density function for the random walk, allowing us to obtain various behaviors for the waiting time and jumping probability distributions. The behavior of these distributions has implications for the diffusion-like equations which emerge from this approach and can be connected to different fractional operators with singular or nonsingular kernels. We also show that diffusion-like equations can exhibit a large class of behaviors related to different processes, particularly anomalous diffusion.On the representation of the Goursat boundary problem solution for the first order partial derivatives stochastic hyperbolic equationshttps://zbmath.org/1526.353442024-02-15T19:53:11.284213Z"Mansimov, Kamil' Baĭramali"https://zbmath.org/authors/?q=ai:mansimov.kamil-bairamali-oglu"Mastaliev, Rashad Ogtaĭ"https://zbmath.org/authors/?q=ai:mastaliev.rashad-ogtai-oglySummary: We study the standard canonical form of a stochastic analog of a system of linear partial differential equations of first order hyperbolic type with Goursat boundary conditions. The stochastic analogue of the Riemann matrix in block form is introduced, an integral representation of the solution of the boundary value problem under consideration is obtained in an explicit integral form in terms of boundary conditions.Stability and moment estimates for the stochastic singular \(\Phi\)-Laplace equationhttps://zbmath.org/1526.353452024-02-15T19:53:11.284213Z"Seib, Florian"https://zbmath.org/authors/?q=ai:seib.florian"Stannat, Wilhelm"https://zbmath.org/authors/?q=ai:stannat.wilhelm"Tölle, Jonas M."https://zbmath.org/authors/?q=ai:tolle.jonas-mSummary: We study stability, long-time behavior and moment estimates for stochastic evolution equations with additive Wiener noise and with singular drift given by a divergence type quasilinear diffusion operator which may not necessarily exhibit a homogeneous diffusivity. Our results cover the singular stochastic \(p\)-Laplace equations and, more generally, singular stochastic \(\Phi\)-Laplace equations with zero Dirichlet boundary conditions. We obtain improved moment estimates and quantitative convergence rates of the ergodic semigroup to the unique invariant measure, classified in a systematic way according to the degree of local degeneracy of the potential at the origin. We obtain new concentration results for the invariant measure and establish maximal dissipativity of the associated Kolmogorov operator. In particular, we recover the results for the curve shortening flow in the plane by \textit{A. Es-Sarhir} et al. [NoDEA, Nonlinear Differ. Equ. Appl. 19, No. 6, 663--675 (2012; Zbl 1266.60117)], and improve the results by \textit{W. Liu} and \textit{J. M. Toelle} [Electron. Commun. Probab. 16, 447--457 (2011; Zbl 1244.60062)].New classes of parabolic pseudo-differential equations, Feller semigroups, contraction semigroups and stochastic process on the \(p\)-adic numbershttps://zbmath.org/1526.353492024-02-15T19:53:11.284213Z"Torresblanca-Badillo, Anselmo"https://zbmath.org/authors/?q=ai:torresblanca-badillo.anselmo"Narváez, Alfredo R. R."https://zbmath.org/authors/?q=ai:narvaez.alfredo-r-r"López-González, José"https://zbmath.org/authors/?q=ai:lopez-gonzalez.joseSummary: Two types of \(p\)-adic pseudo-differential operators (denoted, respectively, by \({\mathcal{T}}_{{\boldsymbol{f}}_1,{\boldsymbol{f}}_2}^l\) and \({\mathcal{J}}_{{\boldsymbol{f}}_1,{\boldsymbol{f}}_2}^{\alpha})\) are introduced in this article. We will show that the operator \({\mathcal{T}}_{{\boldsymbol{f}}_1,{\boldsymbol{f}}_2}^l\) determines certain Feller semigroups and stochastic processes with state space the \(p\)-adic numbers. The second type of these operators (defined on a new class of \(p\)-adic Sobolev space) are connected with contraction semigroups and parabolic pseudo-differential equations.Iterated invariance principle for slowly mixing dynamical systemshttps://zbmath.org/1526.370102024-02-15T19:53:11.284213Z"Galton, Matt"https://zbmath.org/authors/?q=ai:galton.matt"Melbourne, Ian"https://zbmath.org/authors/?q=ai:melbourne.ianSummary: We give sufficient Gordin-type criteria for the iterated (enhanced) weak invariance principle to hold for deterministic dynamical systems. Such an invariance principle is intrinsically related to the interpretation of stochastic integrals. We illustrate this with examples of deterministic fast-slow systems where our iterated invariance principle yields convergence to a stochastic differential equation.Quasimonotone random and stochastic functional differential equations with applicationshttps://zbmath.org/1526.370652024-02-15T19:53:11.284213Z"Bai, Xiaoming"https://zbmath.org/authors/?q=ai:bai.xiaoming"Jiang, Jifa"https://zbmath.org/authors/?q=ai:jiang.jifa"Xu, Tianyuan"https://zbmath.org/authors/?q=ai:xu.tianyuan|xu.tianyuan.1Summary: In this paper, we study monotone properties of random and stochastic functional differential equations and their global dynamics. First, we show that random functional differential equations (RFDEs) generate the random dynamical system (RDS) if and only if all the solutions are globally defined, and establish the comparison theorem for RFDEs and the random Riesz representation theorem. These three results lead to the Borel measurability of coefficient functions in the Riesz representation of variational equations for quasimonotone RFDEs, which paves the way following the Smith line to establish eventual strong monotonicity for the RDS under cooperative and irreducible conditions. Then strong comparison principles, strong sublinearity theorems and the existence of random attractors for RFDEs are proved. Finally, criteria are presented for the existence of a unique random equilibrium and its global stability in the universe of all the tempered random closed sets of the positive cone. Applications to typical random or stochastic delay models in monotone dynamical systems, such as biochemical control circuits, cyclic gene models and Hopfield-type neural networks, are given.Bifurcation theory for SPDEs: finite-time Lyapunov exponents and amplitude equationshttps://zbmath.org/1526.370672024-02-15T19:53:11.284213Z"Blömker, Dirk"https://zbmath.org/authors/?q=ai:blomker.dirk"Neamţu, Alexandra"https://zbmath.org/authors/?q=ai:neamtu.alexandraSummary: We consider a stochastic partial differential equation close to bifurcation of pitchfork type, where a one-dimensional space changes its stability. For finite-time Lyapunov exponents we characterize regions depending on the distance from bifurcation and the noise strength where finite-time Lyapunov exponents are positive and thus detect bifurcations. One technical tool is the reduction of the essential dynamics of the infinite-dimensional stochastic system to a simple ordinary stochastic differential equation, which is valid close to the bifurcation.Convergence rate for degenerate partial and stochastic differential equations via weak Poincaré inequalitieshttps://zbmath.org/1526.370842024-02-15T19:53:11.284213Z"Bertram, Alexander"https://zbmath.org/authors/?q=ai:bertram.alexander"Grothaus, Martin"https://zbmath.org/authors/?q=ai:grothaus.martinSummary: We employ weak hypocoercivity methods to study the long-term behavior of operator semigroups generated by degenerate Kolmogorov operators with variable second-order coefficients, which solve the associated abstract Cauchy problem. We prove essential m-dissipativity of the operator, which extends previous results and is key to the rigorous analysis required. We give estimates for the \(L^2\)-convergence rate by using weak Poincaré inequalities. As an application, we obtain estimates for the (sub-)exponential convergence rate of solutions to the corresponding degenerate Fokker-Planck equations and of weak solutions to the corresponding degenerate stochastic differential equation with multiplicative noise.Periodic measures for the stochastic delay modified Swift-Hohenberg lattice systemshttps://zbmath.org/1526.370862024-02-15T19:53:11.284213Z"Wang, Fengling"https://zbmath.org/authors/?q=ai:wang.fengling"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Li, Yangrong"https://zbmath.org/authors/?q=ai:li.yangrong"Wang, Renhai"https://zbmath.org/authors/?q=ai:wang.renhaiSummary: In this paper, the existence and the limiting behavior of periodic measures for the periodic stochastic modified Swift-Hohenberg lattice systems with variable delays are analyzed. We first prove the existence and uniqueness of global solution when the nonlinear \(\mathcal{T}\)-periodic drift and diffusion terms are locally Lipchitz continuous and linearly growing. Then we show the existence of periodic measures of the system under some assumptions. Finally, by strengthening the assumptions, we prove that the set of all periodic measures is weakly compact, and we also show that every limit point of a sequence of periodic measures of the original system must be a periodic measure of the limiting system when the noise intensity tends to zero.Tauberian Korevaarhttps://zbmath.org/1526.400012024-02-15T19:53:11.284213Z"Bingham, N. H."https://zbmath.org/authors/?q=ai:bingham.nicholas-hSummary: We focus on the Tauberian work for which Jaap Korevaar is best known, together with its connections with probability theory. We begin with a brief sketch of the field up to Beurling's work. We follow with three sections on Beurling aspects: Beurling slow variation; the Beurling Tauberian theorem for which it was developed; Riesz means and Beurling moving averages. We then give three applications from probability theory: extremes, laws of large numbers, and large deviations. We then turn briefly to other areas of Korevaar's work. We close with a personal postscript (whence our title).Convergence of Vilenkin-Fourier series in variable Hardy spaceshttps://zbmath.org/1526.420142024-02-15T19:53:11.284213Z"Weisz, Ferenc"https://zbmath.org/authors/?q=ai:weisz.ferencLet \(p(.):[0,1)\to (0,\infty)\) be a variable exponent function and for any measurable set \(A\subset [0,1)\), let us denote
\[
p_{-}(A)=\operatorname*{essinf}_{x\in A}p(x)\quad \text{and} \quad p_{+}(A)=\operatorname*{esssup}_{x\in A}p(x).
\]
Particularly, if \(A=[0,1)\) then \(p\_(A)\) and \(p_{+}(A)\) are denoted as \(p\-\) and \(p_{+}\). Let us denote by \(\mathcal{P}\) the collection of all variable exponent \(p(.)\), such that, \(0<p\_ \leq p_{+}\). So, the author defines by \(C^{\log}\) the set of all functions \(p(.)\in \mathcal{P}\), satisfying the following property: it exists a positive constant \(C_{\log}(p)\), such that, for any \(x,y\in [0,1)\) we have,
\[
|p(x)-p(y)|\leq \frac{C_{\log}(p)}{\text{log}(e+1/|x-y|)}.
\]
The above inequality is called the log-Hölder continuous condition. Now, let \((p_n)_{n\in\mathbb{N}}\) be a sequence of natural numbers with entries at least 2. Suppose that \((p_n)_{n\in\mathbb{N}}\) is bounded. Then, let us introduce the notations \(P_0=1\) and \(P_{n+1}=\prod_{k=0}^{n}p_k\), \(k\in\mathbb{N}\), and let \(\mathcal{F}_n\) be the \(\sigma\)-algebra defined as
\[
\mathcal{F}_n=\sigma\{[kP^{-1}_n,(k+1)P^{-1}_n):0\leq k\leq P_n\}.
\]
Therefore, the interval of the form \([kP^{-1}_n,(k+1)P^{-1}_n)\) for some \(k,n\in\mathbb{N}\) and \(0\leq k\leq P_n\), is called a Vilenkin interval. An integrable sequence \(f=(f_n)_{n\in\mathbb{N}}\) is a martingale if \(f_n\) is \(\mathcal{F}_n\)-measurable \(\forall n\in\mathbb{N}\) and \(E_nf_m=f_n\) if \(n\leq m\). Hence, a martingale with respect to \((\mathcal{F}_n)_n\) is called a Vilenkin martingale. In this way, the author introduces the maximal function given by \(M(f)=\sup_{n\in\mathbb{N}}|f_n|\) and the variable martingale Hardy spaces are defined as
\[
H_{p(.)}=\{f=(f_n)_{n\in\mathbb{N}}:\|f\|_{H_{p(.)}}=\|M(f)\|_{p(.)}<\infty\},
\]
and
\[
H_{p(.),q}=\{f=(f_n)_{n\in\mathbb{N}}:\|f\|_{H_{p(.),q}}=\|M(f)\|_{p(.),q}<\infty\}.
\]
Now, every point \(x\in[0,1)\) can be written as \(x=\sum_{k=0}^\infty \frac{x_k}{P_{k+1}}\), \(0\leq x_k\leq p_k\), \(x_k\in \mathbb{N}\) and the functions \(r_n(x)=\exp(2\pi i/p_n)\), \(n\in \mathbb{N}\), are called generalized Rademacher functions. Therefore, the functions \(w_n(x)=\prod_{k=0}^\infty r_k(x)^{n_k}\) constitute a Vilenkin system, where \(n=\sum_{k=0}^{\infty}n_kP_k\) and \(0\leq n_k\leq p_k\). The Vilenkin Dirichlet kernels are defined as \(D_n=\sum_{k=0}^{n-1}w_k\). Also, if \(f\in L_1\) then \(\hat{f}(n)=\int_0^1 f\bar{w_n}d\lambda\), \(n\in\mathbb{N}\), is the Vilenkin-Fourier coefficient and if \(f=(f_n)_{n\in\mathbb{N}}\) is a martingale, then \(\hat{f}(n)=\lim_{k\to\infty}\int_0^1 f_k\bar{w_n}d\lambda\). The \(n\)th partial sum of the Vilenkin Fourier series of a martingale \(f\) is given as
\[
S_nf=\sum_{k=0}^{\infty}\hat{f}(k)w_k.
\]
In this context, the author obtains the following result. Let \(p(.)\in C^{\log}\) with \(1<p\_\leq p_{+}<\infty\). If \(f\in L_{p(.)}\) then \(\sup_{n\in\mathbb{N}}\|S_n f\|_{p(.)}\lesssim \|f\|_{p(.)}\). Moreover, \(\lim_{n\to\infty}S_nf=f\) in the \(L_{p(.)}\)-norm.
Finally, let us consider the Fejer mean of order \(n\) of the Vilenkin-Fourier series of \(f\), as \(\sigma_nf=\frac{1}{n}\sum_{k=1}^{n}S_kf\) and the maximal operators \(\sigma_*f=\operatorname*{sup}_{n\in\mathbb{N}}|\sigma_nf|\). Then, through atomic decompositions, the author gets the following results, which we summarize as follows:
Theorem 1.
Let \(p(.)\in C^{\log}\) satisfy \(\frac{1}{p\-}-\frac{1}{p_{+}}<1\). If \(\frac{1}{2}<p\_<\infty\), then \(\| \sigma_*f\|_{p(.)}\lesssim \|f\|_{H_{p(.)}}\), for \(f\in H_{p(.)}\).
Theorem 2.
Let \(p(.)\in C^{\log}\) satisfy \(\frac{1}{p\-}-\frac{1}{p_{+}}<1\). If \(0<q\leq \infty\), then
\[
\|\operatorname*{sup}_{n\in\mathbb{N}}|\sigma_{P_n}f| \|_{L_{p(.)},q}\lesssim \|f\|_{H_{p(.)},q},
\ \text{for}
\ f\in H_{p(.),q}
\]
and \(\| \sigma_*f\|_{p(.),q}\lesssim \|f\|_{H_{p(.)},q}\), for \(f\in H_{p(.),q}\).
Corollary 1.
Let \(p(.)\in C^{\log}\) satisfy \(\frac{1}{p\-}-\frac{1}{p_{+}}<1\) and \(0<q\leq \infty\). If \(f\in H_{p(.),q}\) then \(\sigma_{P_n}f\) converges almost everywhere on \([0,1)\) and in the \(L_{p(.),q}\) norm. If in addition, \(\frac{1}{2}<p\_<\infty\), then the convergence holds for \(\sigma_n f\) too.
Corollary 2.
Let \(p(.)\in C^{\log}\) satisfy \(\frac{1}{p\-}-\frac{1}{p_{+}}<1\), \(1\leq p\_<\infty\), \(0<q\leq \infty\) and \(f\in H_{p(.),q}\). Then, \(\lim_{n\to\infty} \sigma_nf(x)=f(x)\), for a.e. \(x\in[0,1)\) and in the \(L_{p(.),q}\) norm.
Corollary 3.
If \(f\in L_1\), then \(\lim_{n\to\infty}\sigma_nf(x)=f(x)\) for a.e. \(x\in [0,1)\).
Reviewer: Iris Athamaica López Palacios (Caracas)Expanding the Fourier transform of the scaled circular Jacobi \(\beta\) ensemble densityhttps://zbmath.org/1526.420162024-02-15T19:53:11.284213Z"Forrester, Peter J."https://zbmath.org/authors/?q=ai:forrester.peter-j"Shen, Bo-Jian"https://zbmath.org/authors/?q=ai:shen.bo-jianSummary: The family of circular Jacobi \(\beta\) ensembles has a singularity of a type associated with \textit{M. E. Fisher} and \textit{R. E. Hartwig} [Adv. Chem. Phys. 15, 333--353 (1968; \url{doi.org/10.1002/9780470143605.ch18})] in the theory of Toeplitz determinants. Our interest is in the Fourier transform of the corresponding \(N \rightarrow \infty\) bulk scaled spectral density about this singularity, expanded as a series in the Fourier variable. Various integrability aspects of the circular Jacobi \(\beta\) ensemble are used for this purpose. These include linear differential equations satisfied by the scaled spectral density for \(\beta = 2\) and \(\beta = 4\), and the loop equation hierarchy. The polynomials in the variable \(u=2/\beta\) which occur in the expansion coefficents are found to have special properties analogous to those known for the structure function of the circular \(\beta\) ensemble, specifically in relation to the zeros lying on the unit circle \(|u|=1\) and interlacing. Comparison is also made with known results for the expanded Fourier transform of the density about a guest charge in the two-dimensional one-component plasma.Polynomially growing harmonic functions on connected groupshttps://zbmath.org/1526.430072024-02-15T19:53:11.284213Z"Perl, Idan"https://zbmath.org/authors/?q=ai:perl.idan"Yadin, Ariel"https://zbmath.org/authors/?q=ai:yadin.arielSummary: We study the connection between the dimension of certain spaces of harmonic functions on a group and its geometric and algebraic properties.
Our main result shows that (for sufficiently ``nice'' random walk measures) a connected, compactly generated, locally compact group has polynomial volume growth if and only if the space of linear growth harmonic functions has finite dimension.
This characterization is interesting in light of the fact that Gromov's theorem regarding finitely generated groups of polynomial growth does not have an analog in the connected case. That is, there are examples of connected groups of polynomial growth that are not nilpotent by compact. Also, the analogous result for the discrete case has only been established for solvable groups and is still open for general finitely generated groups.Real interpolation of variable martingale Hardy spaces and BMO spaceshttps://zbmath.org/1526.460192024-02-15T19:53:11.284213Z"Lu, Jianzhong"https://zbmath.org/authors/?q=ai:lu.jianzhong"Weisz, Ferenc"https://zbmath.org/authors/?q=ai:weisz.ferenc"Zhou, Dejian"https://zbmath.org/authors/?q=ai:zhou.dejianSummary: In this paper, we mainly consider the real interpolation spaces for variable Lebesgue spaces defined by the decreasing rearrangement function and for the corresponding martingale Hardy spaces. Let \(0<q\leq \infty\) and \(0<\theta <1\). Our three main results are the following:
\[
\begin{aligned} ({\mathcal{L}}_{p(\cdot)}({\mathbb{R}}^n),L_{\infty}({\mathbb{R}}^n))_{\theta, q} &= {\mathcal{L}}_{{p(\cdot)}/(1-\theta),q}({\mathbb{R}}^n),\\
({\mathcal{H}}_{p(\cdot)}^s(\Omega),H_{\infty}^s(\Omega))_{\theta,q} &= {\mathcal{H}}_{{p(\cdot)}/(1-\theta),q}^s(\Omega) \end{aligned}
\]
and
\[
\begin{aligned} ({\mathcal{H}}_{p(\cdot)}^s(\Omega), \mathrm{BMO}_2(\Omega))_{\theta, q} = {\mathcal{H}}_{{p(\cdot)}/(1- \theta),q}^s(\Omega), \end{aligned}
\]
where the variable exponent \(p(\cdot)\) is a measurable function.Measure theory, probability, and stochastic processeshttps://zbmath.org/1526.600012024-02-15T19:53:11.284213Z"Le Gall, Jean-François"https://zbmath.org/authors/?q=ai:le-gall.jean-francoisThis book covers probability theory and stochastic processes at a graduate level. It begins with measure theory, allowing readers with a background in measure theory to skip ahead and start from Part II, which focuses on probability theory. Part II covers most topics commonly found in other graduate-level probability books. Part III, stochastic processes, delves into martingale theory, Markov chains, and Brownian motions. However, it's worth noting that while this book covers Brownian motion, it does not delve into the general theory of Markov processes with uncountable state spaces and continuous-time martingales, which is based on more advanced functional analysis. For a comprehensive treatment of these advanced topics, one may refer to [Markov processes. Characterization and convergence. Hoboken, NJ: John Wiley \& Sons (2005; Zbl 1089.60005)] by \textit{S. N. Ethier} and \textit{T. G. Kurtz}. Additionally, this book provides a set of good exercise problems at the end of each chapter, although the number of problems is relatively small. Many topics covered in Part III have applications in finance, but it's essential to understand that this book focuses solely on theoretical aspects and does not address these practical applications. In conclusion, this book is exceptionally well written in a concise manner and is suitable for individuals with a strong background in undergraduate real analysis and undergraduate probability.
Reviewer: Eunghyun Lee (Astana)Mechanistic Markov models for the evolution of gene families. (Abstract of thesis)https://zbmath.org/1526.600022024-02-15T19:53:11.284213Z"Diao, Jiahao"https://zbmath.org/authors/?q=ai:diao.jiahao(no abstract)Exact-size sampling of enriched trees in linear timehttps://zbmath.org/1526.600032024-02-15T19:53:11.284213Z"Panagiotou, Konstantinos"https://zbmath.org/authors/?q=ai:panagiotou.konstantinos-d"Ramzews, Leon"https://zbmath.org/authors/?q=ai:ramzews.leon"Stufler, Benedikt"https://zbmath.org/authors/?q=ai:stufler.benediktSummary: We create a novel connection between Boltzmann sampling methods and Devroye's algorithm to develop highly efficient sampling procedures that generate objects from important combinatorial classes with a given size \(n\) in expected time \(O(n)\). This performance is best possible and significantly improves the state of the art for samplers of subcritical graph classes (such as cactus graphs, outerplanar graphs, and series-parallel graphs), subcritical substitution-closed classes of permutations, Bienaymé-Galton-Watson trees conditioned on their number of leaves, and several further examples. Our approach allows for this high level of universality, as it applies in general to classes admitting bijective encodings by so-called enriched trees, which are rooted trees with additional structures on the offspring of each node.Conditional probability on full Łukasiewicz tribeshttps://zbmath.org/1526.600042024-02-15T19:53:11.284213Z"Eliaš, Peter"https://zbmath.org/authors/?q=ai:elias.peter"Frič, Roman"https://zbmath.org/authors/?q=ai:fric.romanIn the paper, conditional probability and stochastic dependence/independence is studied on a full Łukasiewicz tribe of all measurable functions from a measurable space into [0,1]. The results are based on properties of joint experiments and the notion of stochastic channel, a construct equivalent to the notion of Markov kernel between two measurable spaces. The paper is a categorical view on probability theory, dependence/independence and conditional probability on full Łukasiewicz tribes.
Reviewer: Martin Kalina (Bratislava)Strong Shannon-McMillan-Breiman's theorem for locally compact groupshttps://zbmath.org/1526.600052024-02-15T19:53:11.284213Z"Forghani, Behrang"https://zbmath.org/authors/?q=ai:forghani.behrang"Nguyen, May"https://zbmath.org/authors/?q=ai:nguyen.maySummary: We prove that for a vast class of random walks on a compactly generated group, the exponential growth of convolutions of a probability density function along almost every sample path is bounded by the growth of the group. As an application, we show that the almost sure and \(L^1\) convergences of the Shannon-McMillan-Breiman theorem hold for compactly supported random walks on compactly generated groups with subexponential growth.Upper bounds for the maximum deviation of the Pearcey processhttps://zbmath.org/1526.600062024-02-15T19:53:11.284213Z"Charlier, Christophe"https://zbmath.org/authors/?q=ai:charlier.christopheSummary: The Pearcey process is a universal point process in random matrix theory and depends on a parameter \(\rho\in\mathbb{R} \). Let \(N(x)\) be the random variable that counts the number of points in this process that fall in the interval \([-x,x]\). In this note, we establish the following global rigidity upper bound:
\[
\lim_{s \to \infty}\mathbb{P}\left( \sup_{x > s} \left| \frac{N (x) - \left( \frac{3 \sqrt{3}}{4 \pi} x^{\frac{4}{3}} - \frac{\sqrt{3} \rho}{2 \pi} x^{\frac{2}{3}}\right)}{\log x}\right| \leq \frac{4 \sqrt{2}}{3 \pi} + \epsilon\right)=1,
\]
where \(\epsilon>0\) is arbitrary. We also obtain a similar upper bound for the maximum deviation of the points, and a central limit theorem for the individual fluctuations. The proof is short and combines a recent result of Dai, Xu and Zhang with another result of Charlier and Claeys.Randomly coupled differential equations with elliptic correlationshttps://zbmath.org/1526.600072024-02-15T19:53:11.284213Z"Erdős, László"https://zbmath.org/authors/?q=ai:erdos.laszlo"Krüger, Torben"https://zbmath.org/authors/?q=ai:kruger.torben"Renfrew, David"https://zbmath.org/authors/?q=ai:renfrew.davidIn order to describe the evolution of a network of \(N\) fully connected neurons, one can consider the following system of linear differential equations:
\[
\partial_t u(t) = -u(t) + g X u(t),
\]
where \(u\) has values in \(\mathbb{C}^N\), \(X\in\mathbb{C}^{N\times N}\) is a random matrix, and \(g>0\) is a coupling parameter.
In the simplest setting the entries of the matrix \(X\) would be independent or even i.i.d.\ random variables. However, recent experimental data suggests that in reality \(X_{ij}\) and \(X_{ij}\) are not independent. One could therefore assume that for every \(i\neq j\),
\[
\mathbb{E} X_{ij} X_{ji} = \rho \mathbb{E} |X_{ij}|^2 = \frac{\rho}{N},
\]
where the complex number \(\rho\) is the correlation coefficient (the pairs \((X_{ij}, X_{ji})\) are still assumed to be independent for different index pairs \(\{i,j\}\)). Note that \(\rho=0\) in the independent case and \(\rho = 1\) if \(X=X^*\) is Hermitian.
Depending on the value of the coupling parameter \(g\) the solution \(u\) typically grows or decays exponentially, but for a critical value it exhibits a power law decay with an exponent which depends on the symmetry properties of \(X\). The extreme cases \(\rho=0\) and \(\rho = 1\) were studied in a previous article of the authors [SIAM J. Math. Anal. 50, No. 3, 3271--3290 (2018; Zbl 1392.60011)]. In the paper under review, they focus on the remaining case \(0<|\rho| <1\) (some mathematically non-rigorous analysis of this intermediate case appears in the literature; here the authors present complete proofs). The results about the long time asymptotic behaviour of the above system of \(N\) linear differential equations are presented under a general assumption that \(X\) is an elliptic or elliptic-type random matrix.
One of the main tools in the proofs is an asymptotically precise formula for \(\frac{1}{N}\mathbb{E} \operatorname{Tr}(f(X) g(X^*))\), where \(f\) and \(g\) are analytic functions. The paper contains 4 figures.
Reviewer: Michał Strzelecki (Warszawa)Approximation of beta-Jacobi ensembles by beta-Laguerre ensembleshttps://zbmath.org/1526.600082024-02-15T19:53:11.284213Z"Ma, Yutao"https://zbmath.org/authors/?q=ai:ma.yutao"Shen, Xinmei"https://zbmath.org/authors/?q=ai:shen.xinmeiSummary: Consider beta-Laguerre ensembles \(\boldsymbol{\mu}\) with parameters \(m\), \( a_1\) and beta-Jacobi ensembles \(\boldsymbol{\lambda}\) with parameters \(m\), \(a_1\), \( a_2\). With the help of tridiagonal models of beta ensembles, we are able to prove that \(\lim_{a_2 \to \infty} d(\mathcal{L} (2a \boldsymbol{\lambda}), \mathcal{L} (\boldsymbol{\mu}))=0\) if \(a_1 m=o(a_2)\) and \(\underline{\lim}_{a_2 \to \infty} d (\mathcal{L} (2a\boldsymbol{\lambda}), \mathcal{L} (\boldsymbol{\mu})) > 0\) if \(\lim_{a_2 \to \infty} \frac{a_1 m}{a_2} =\sigma > 0\), by contrast, where \(a:= a_1 + a_2\) and \(d\) is total variation distance or Kullback-Leibler divergence. This result improves the approximation in [\textit{T. Jiang}, Bernoulli 19, No. 3, 1028--1046 (2013; Zbl 1278.60013)].Edge behavior of higher complex-dimensional determinantal point processeshttps://zbmath.org/1526.600092024-02-15T19:53:11.284213Z"Molag, L. D."https://zbmath.org/authors/?q=ai:molag.leslie-dSummary: As recently proved in generality by Hedenmalm and Wennman, it is a universal behavior of complex random normal matrix models that one finds a complementary error function behavior at the boundary (also called edge) of the droplet as the matrix size increases. Such behavior is seen both in the density of the eigenvalues and the correlation kernel, where the Faddeeva plasma kernel emerges. These results are neatly expressed with the help of the outward unit normal vector on the edge. We prove that such universal behaviors transcend this class of random normal matrices, being also valid in a specific ``elliptic'' class of determinantal point processes defined on \(\mathbb{C}^d\), which are higher dimensional generalizations of the determinantal point processes describing the eigenvalues of the complex Ginibre ensemble and the complex elliptic Ginibre ensemble. These models describe a system of particles in \(\mathbb{C}^d\) with mutual repulsion, that are confined to the origin by an external field \(\mathscr{V}(z) = |z|^2 - \tau \Re(z_1^2 + \dots + z_d^2)\), where \(0 \leq \tau < 1\). Their average density of particles converges to a uniform law on a \(2d\)-dimensional ellipsoidal region. It is on the hyperellipsoid bounding this region that we find a complementary error function behavior and the Faddeeva plasma kernel. To the best of our knowledge, this is the first instance of the Faddeeva plasma kernel emerging in a higher dimensional model. The results provide evidence for a possible edge universality theorem for determinantal point processes on \(\mathbb{C}^d\).Noncentral complex Wishart matrices: moments and correlation of minorshttps://zbmath.org/1526.600102024-02-15T19:53:11.284213Z"Tralli, Velio"https://zbmath.org/authors/?q=ai:tralli.velio"Conti, Andrea"https://zbmath.org/authors/?q=ai:conti.andrea.1Summary: Complex Wishart matrices are a class of random matrices with numerous emerging applications. In particular, the statistical characterization of such class of random matrices is essential for solving problems in various fields, including statistics, finance, physics and engineering. This paper establishes a new way to solve such problems based on the statistical moments and correlation of the minors. The first two minors' moments and correlation for noncentral complex Wishart matrices are derived as a function of size and degrees of freedom, as well as of covariance and noncentrality matrices. As a case study, the findings are applied to the statistical characterization of the capacity of wireless multiple-input-multiple-output systems.Sample canonical correlation coefficients of high-dimensional random vectors: local law and Tracy-Widom limithttps://zbmath.org/1526.600112024-02-15T19:53:11.284213Z"Yang, Fan"https://zbmath.org/authors/?q=ai:yang.fan.33|yang.fan.14|yang.fan.1|yang.fan.28|yang.fan.2Summary: Consider two random vectors \(\mathbf{C}_1^{1 / 2}\mathbf{x}\in \mathbb{R}^p\) and \(\mathbf{C}_2^{1 / 2}\mathbf{y}\in \mathbb{R}^q\), where the entries of \(\mathbf{x}\) and \(\mathbf{y}\) are i.i.d. random variables with mean zero and variance one, and \(\mathbf{C}_1\) and \(\mathbf{C}_2\) are respectively, \(p\times p\) and \(q\times q\) deterministic population covariance matrices. With \(n\) independent samples of \(( \mathbf{C}_1^{1 / 2}\mathbf{x}, \mathbf{C}_2^{1 / 2}\mathbf{y})\), we study the sample correlation between these two vectors using canonical correlation analysis. Under the high-dimensional setting with \(p/n\to c_1\in(0,1)\) and \(q/n\to c_2\in(0,1- c_1)\) as \(n\to\infty \), we prove that the largest sample canonical correlation coefficient converges to the Tracy-Widom distribution as long as we have \(\lim_{s \to \infty} s^4\mathbb{P}(| x_{i j}|\geq s)=0\) and \(\lim_{s \to \infty} s^4\mathbb{P}(| y_{i j}|\geq s)=0\), which we believe to be a sharp moment condition. This extends the result in [\textit{X. Han} et al., Bernoulli 24, No. 4B, 3447--3468 (2018; Zbl 1407.60008)], which established the Tracy-Widom limit under the assumption that all moments exist for the entries of \(\mathbf{x}\) and \(\mathbf{y}\). Our proof is based on a new linearization method, which reduces the problem to the study of a \((p+q+2n)\times (p+q+2n)\) random matrix \(H\). In particular, we shall prove an optimal local law on its inverse \(G:= H^{- 1} \), called resolvent. This local law is the main tool for both the proof of the Tracy-Widom law in this paper, and the study in [\textit{F. Yang}, Electron. J. Probab. 27, Paper No. 86, 71 p. (2022; Zbl 1498.60102); \textit{Z. Ma} and \textit{F. Yang}, Bernoulli 29, No. 3, 1905--1932 (2023; Zbl 1518.15042)] on the canonical correlation coefficients of high-dimensional random vectors with finite rank correlations.Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walkshttps://zbmath.org/1526.600122024-02-15T19:53:11.284213Z"Kabluchko, Zakhar"https://zbmath.org/authors/?q=ai:kabluchko.zakhar-a"Marynych, Alexander"https://zbmath.org/authors/?q=ai:marynych.alexander-vThe Lah number \(L(n,k)\) is the number of ways to partition the set \(\{1,2,\dots,n\}\) into \(k\) nonempty tuples (i.e., linearly ordered sets). It is known that
\[
L(n,k)=\sum_{j=k}^n \mathfrak{s}(n,j) S(j,k) = \frac{n!}{k!} \binom{n-1}{k-1}, \;\; 1 \leq k \leq n,
\]
where \(\mathfrak{s}(n,j)\) is the unsigned Stirling number of the first kind and \(S(j,k)\) is the Stirling number of the second kind.
This paper defines Lah distribution with parameters \(n \in \mathbb{N}\) and \(k \in \{1,2,\dots,n\}\) as following
\[
\mathbb{P}(X=j) = \frac{\mathfrak{s}(n,j) S(j,k)}{L(n,k)}, \;\; k \leq j \leq n.
\]
The authors provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. The limit theorems which are proved in this paper for the Lah distribution imply the neighborliness properties of \(C_{n,d}\), where \(C_{n,d}\) is the convex null of some random walks.
Reviewer: Ping Sun (Shenyang)Does testing positive for COVID-19, mean being infectedhttps://zbmath.org/1526.600132024-02-15T19:53:11.284213Z"Rigny, Agnès"https://zbmath.org/authors/?q=ai:rigny.agnesThis article explains conditional probability and how to estimate the infection rate of a population using tests.
Reviewer: Franz Lemmermeyer (Jagstzell)Second-order properties for planar Mondrian tessellationshttps://zbmath.org/1526.600142024-02-15T19:53:11.284213Z"Betken, Carina"https://zbmath.org/authors/?q=ai:betken.carina"Kaufmann, Tom"https://zbmath.org/authors/?q=ai:kaufmann.tom"Meier, Kathrin"https://zbmath.org/authors/?q=ai:meier.kathrin"Thäle, Christoph"https://zbmath.org/authors/?q=ai:thale.christophAuthors' abstract: In this paper planar STIT tessellations with weighted axis-parallel cutting directions are considered. They are known also as weighted planar Mondrian tessellations in the machine learning literature, where they are used in random forest learning and kernel methods. Various second-order properties of such random tessellations are derived, in particular, explicit formulas are obtained for suitably adapted versions of the pair- and cross-correlation functions of the length measure on the edge skeleton and the vertex point process. Also, explicit formulas and the asymptotic behaviour of variances are discussed in detail.
Reviewer: Christian Richter (Jena)Investigation of convex bodies in \(\mathbb R^3\) by support planeshttps://zbmath.org/1526.600152024-02-15T19:53:11.284213Z"Harutyunyan, H. O."https://zbmath.org/authors/?q=ai:harutyunyan.h-o"Ohanyan, V. K."https://zbmath.org/authors/?q=ai:ohanyan.v-kSummary: Let \(\mathbb{R}^3\) be the 3-dimensional Euclidean space and \({\mathbf{D}}\) be a bounded convex body \(D\subset\mathbb{R}^3 \). Consider a family of support planes for which \({\mathbf{D}}\) is an envelope. How can we obtain information about \(D\) from the support planes? Conditions under which a given convex body is the envelope of a family of planes are obtained. Therefore the distances of these planes from the origin will be the support functions of this body \(D\). In particular, we have cited expressions for the surface area and the volume of body \({\mathbf{D}}\) in terms of support planes.Geodesic stars in random geometryhttps://zbmath.org/1526.600162024-02-15T19:53:11.284213Z"Le Gall, Jean-François"https://zbmath.org/authors/?q=ai:le-gall.jean-francoisA geodesic in a metric space \((E, d)\) is a continuous path \((\gamma(t))_{t\in[0,\delta]}\), where \(\delta > 0\), such that \(d(\gamma(s), \gamma(t)) = |s - t|\) for every \(s, t \in [0, \delta]\). For every \(t \in(0, \delta)\), \(\gamma(t)\) is an interior point of the geodesic (whereas \(\gamma(0)\) and \(\gamma(\delta)\) are its endpoints). If \(m\ge 1\) is an integer, then a point \(x\) is a geodesic star with \(m\) arms (in short, an \(m\)-geodesic star) if there exist \(\delta > 0\) and \(m\) geodesics \((\gamma_1(t))_{t\in[0,\delta]}, \dots, (\gamma_m(t))_{t\in[0,\delta]}\) such that \(\gamma_1(0) = \gamma_2(0) = \dots = \gamma_m(0) = x\) and the sets \(\{\gamma_j(t): t \in (0, \delta]\}\), for \(j \in \{1, \dots, m\}\), are disjoint. If \((E, d)\) is a geodesic space, any pair of distinct points is connected by a (possibly not unique) geodesic, and it is then immediate that every point is a \(1\)-geodesic star.
The main result of the paper is the following statement.
Theorem 1. Let \((m_{\infty}, D)\) denote the Brownian sphere. For every integer \(m\in \{1, 2, 3, 4\}\), let \({\mathcal{E}}_m\) be the set of all \(m\)-geodesic stars in \((m_\infty, D)\). Then, the Hausdorff dimension of \({\mathcal{E}}_m\) is a.s. equal to \(5-m\).
The upper bound \(\dim({\mathcal{E}}_m) \leq 5 - m\) has been obtained in [\textit{J. Miller} and \textit{W. Qian}, ``Geodesics in the Brownian map: Strong confluence and geometric structure'', Preprint, \url{arXiv:2008.02242}]. So the contribution of the present work is to prove the corresponding lower bound. Note that \(m\)-geodesic stars in the Brownian sphere were first discussed in [\textit{G. Miermont}, Acta Math. 210, No. 2, 319--401 (2013; Zbl 1278.60124)]. The Brownian sphere is a geodesic space, and thus \({\mathcal{E}}_1 = m_{\infty}\) so that, in the case \(m = 1\), the result follows from the known fact [\textit{J.-F. Le Gall}, Invent. Math. 169, No. 3, 621--670 (2007; Zbl 1132.60013)] that \(\dim(m_{\infty}) = 4\). Any interior point of a geodesic is a \(2\)-geodesic star, and, therefore, \({\mathcal{E}}_2\) contains the set of all interior points of all geodesics. However, the authors of [Miller and Qian, loc. cit.] proved that the Hausdorff dimension of the latter set is 1 (it is obviously greater than or equal to 1), thus confirming a conjecture from [\textit{O. Angel} et al., Ann. Probab. 45, No. 5, 3451--3479 (2017; Zbl 1407.60018)]. Since \(\dim({\mathcal{E}}_2) =3\), this implies, that typical \(2\)-geodesic stars are not interior points of geodesics.
\textbf{Open Problem.} Prove or disprove the existence of \(5\)-geodesic stars in the Brownian map.
The paper is organized as follows. Section 2 is devoted to a number of preliminaries, including the Brownian snake construction of the Brownian sphere as a measure metric space with two distinguished points denoted by \(x_*\) and \(x_0\), and a discussion of the symmetry properties of the Brownian sphere, which, roughly speaking, say that \(x_*\) and \(x_0\) play the same role as two points chosen independently according to the (normalized) volume measure. Theorem 8 of Section 3 shows that the hull of radius \(r>0\) centered at \(x_*\) (relative to \(x_0\)) is independent of its complement conditionally on its boundary size, and the complement itself is a Brownian disk; this is, in fact, an analog of a result proved in [\textit{J.-F. Le Gall} and \textit{A. Riera}, Probab. Theory Relat. Fields 181, No. 1--3, 571--645 (2021; Zbl 1480.60023)] for the Brownian plane. An important notion of Section 4 is a slice which separates two successive disjoint geodesics from the hull boundary to the ball of radius \(\varepsilon\). Section 5 uses the results of Section 3 to derive the key estimate in Lemma 15. Section 6 then gives the proof of Theorem 1 along the lines of the preceding discussion. The Appendix contains the proofs of a couple of technical lemmas, including the strong coupling between the Brownian plane and the Brownian sphere that is used to justify the zero-one law argument.
Reviewer: Viktor Ohanyan (Erevan)Hellinger's distance and correlation for a subclass of stable distributionshttps://zbmath.org/1526.600172024-02-15T19:53:11.284213Z"Mesropyan, M. T."https://zbmath.org/authors/?q=ai:mesropyan.m-t"Bardakhchyan, V. G."https://zbmath.org/authors/?q=ai:bardakhchyan.vardan-gSummary: We investigated correlation retrieval procedure from Hellinger's distance. We found monotone relation of Hellinger's distance and positive correlation in a subclass of stable distributed random variables, with \(\alpha>1\) and \(\mu=\beta=0\). We implemented a technique suitable for the class of stable distributions, and showed that this positive relation holds even for the case of Levy distribution, i.e., \( \alpha=1/2, \beta=1\), and \(\mu=0\).Characterizations of stochastic ordering for non-negative random variableshttps://zbmath.org/1526.600182024-02-15T19:53:11.284213Z"Yao, Jia"https://zbmath.org/authors/?q=ai:yao.jia"Lou, Bendong"https://zbmath.org/authors/?q=ai:lou.bendong"Pan, Xiaoqing"https://zbmath.org/authors/?q=ai:pan.xiaoqingThe authors establish the following characterization of the usual stochastic order between random variables \(X\) and \(Y\) supported on \([0,\infty)\) with density functions \(f\) and \(g\), respectively: Suppose \(l(x)=\log(f(x)/g(x))\) is a continuous function with at most two roots, such that \(l(x)<0\) for all \(x\) greater than the largest such root. Then \(X\) is stochastically smaller than \(Y\) if and only if \(l(0)\geq0\). Analogous characterizations are also established for the likelihood ratio order, the hazard rate order, and the reversed hazard rate order. The conditions in these characterizations are weaker than the concavity of \(l(x)\) required in the previous work of \textit{Y. Yu} [J. Appl. Probab. 46, 244--254 (2009; Zbl 1161.60308)].
Reviewer: Fraser Daly (Edinburgh)Properties of marginal sequential Monte Carlo methodshttps://zbmath.org/1526.600192024-02-15T19:53:11.284213Z"Crucinio, Francesca R."https://zbmath.org/authors/?q=ai:crucinio.francesca-romana"Johansen, Adam M."https://zbmath.org/authors/?q=ai:johansen.adam-mSummary: We provide a framework which admits a number of ``marginal'' sequential Monte Carlo (SMC) algorithms as particular cases -- including the marginal particle filter [\textit{M. Klaas} et al., ``Toward practical \(N^2\) Monte Carlo: the marginal particle filter'', Preprint, \url{arXiv:1207.1396}], the independent particle filter [\textit{M. T. Lin} et al., J. Am. Stat. Assoc. 100, No. 472, 1412--1421 (2005; Zbl 1117.62388)] and linear-cost Approximate Bayesian Computation SMC [\textit{S. A. Sisson} et al., Proc. Natl. Acad. Sci. USA 104, No. 6, 1760--1765 (2007; Zbl 1160.65005)]. We provide conditions under which such algorithms obey laws of large numbers and central limit theorems and provide some further asymptotic characterizations. Finally, it is shown that the asymptotic variance of a class of estimators associated with certain marginal SMC algorithms is never greater than that of the estimators provided by a standard SMC algorithm using the same proposal distributions.On mixing conditions in proving the asymptotical normality for harmonic crystalshttps://zbmath.org/1526.600202024-02-15T19:53:11.284213Z"Dudnikova, T. V."https://zbmath.org/authors/?q=ai:dudnikova.tatyana-vThe author establishes a central limit theorem theorem for the dynamics of a harmonic crystal in \(\mathbb{R}^d\) with \(n\) components. This central limit theorem establishes weak convergence under \(\alpha\)-mixing conditions of Rosenblatt type on the initial data, a weakening of the \(\varphi\)-mixing conditions used in previous such results. Two proofs of this central limit theorem are given: the first uses the Bernstein method, and the second the Stein-Bolthausen method.
Reviewer: Fraser Daly (Edinburgh)Spectrum of SYK model. II: Central limit theoremhttps://zbmath.org/1526.600212024-02-15T19:53:11.284213Z"Feng, Renjie"https://zbmath.org/authors/?q=ai:feng.renjie"Tian, Gang"https://zbmath.org/authors/?q=ai:tian.gang"Wei, Dongyi"https://zbmath.org/authors/?q=ai:wei.dongyiThe SYK model is a random matrix of the form
\[
H=i^{[q_n/2]}\frac{1}{\sqrt{\binom{n}{q_n}}}\sum_{1\leq i_1<\cdots<i_{q_n}\leq n}J_{i_1\cdots i_{q_n}}\psi_{i_1}\cdots\psi_{i_{q_n}}\,,
\]
where \(n\) is an even integer, the \(J_{i_1\cdots i_{q_n}}\) are independent and identically distributed random variables with zero mean and unit variance, and the \(\psi_j\) are Majorana fermions which can be represented as \(L_n\times L_n\) Hermitian matrices, where \(L_n=2^{n/2}\). Letting \(\lambda_1,\ldots,\lambda_{L_n}\) denote the eigenvalues of \(H\), the authors establish a central limit theorem for statistics of the form
\[
\frac{1}{L_n}\sum_{j=1}^{L_n}f(\lambda_j)\,,
\]
where \(f\) is a real polynomial in the general case, but in the special case where the \(J_{i_1\cdots i_{q_n}}\) are Gaussian may be from a wider class of functions. This central limit theorem is established under the assumption that the \(k\)th moment of \(|J_{i_1\cdots i_{q_n}}|\) is uniformly bounded for any fixed \(k\).
For Part I, see [\textit{R. Feng} et al., Peking Math. J. 2, No. 1, 41--70 (2019; Zbl 1431.83091)]; for Part III, see [\textit{R. Feng} et al., Random Matrices Theory Appl. 9, No. 2, Article ID 2050001, 24 p. (2020; Zbl 1434.60026)].
Reviewer: Fraser Daly (Edinburgh)Another note on the inequality between geometric and \(p\)-generalized arithmetic meanhttps://zbmath.org/1526.600222024-02-15T19:53:11.284213Z"Thäle, Christoph"https://zbmath.org/authors/?q=ai:thale.christophAuthor's abstract: A central limit theorem and a moderate deviations principle for the ratio of geometric and \(p\)-generalized arithmetic mean are shown. Also a Berry-Esseen-type upper bound on the rate of convergence in the central limit theorem is proved. This has implications on the probabilistic version of the question of whether the inequality between geometric and \(p\)-generalized arithmetic mean can be reversed or improved up to multiplicative constants. The involved random vectors we study belong to a class of distributions on the \(\ell_p^n\)-ball introduced by Barthe, Guédon, Mendelson and Naor. The results complement previous limit theorems of Kabluchko, Prochno and Vysotsky.
Reviewer: Edward Omey (Brussel)On the small noise limit in the Smoluchowski-Kramers approximation of nonlinear wave equations with variable frictionhttps://zbmath.org/1526.600232024-02-15T19:53:11.284213Z"Cerrai, Sandra"https://zbmath.org/authors/?q=ai:cerrai.sandra"Xie, Mengzi"https://zbmath.org/authors/?q=ai:xie.mengziThe authors deal with the Smoluchowski-Kramers approximation theorem in several publications. In this article, this theorem is proved for wave equations with little noise for the case of variable friction, non-Lipschitz nonlinear term and unbounded diffusion.
Reviewer: Rózsa Horváth-Bokor (Budakalász)On the probabilities of large deviations of combinatorial sums of independent random variables that satisfy the Linnik conditionhttps://zbmath.org/1526.600242024-02-15T19:53:11.284213Z"Frolov, A. N."https://zbmath.org/authors/?q=ai:frolov.andrei-nikolaevich.1Let \((\{X_{nij}:1\leq i,j\leq n\})_{n=2,3,\ldots}\) be a sequence of matrices of independent random variables such that \(\sum_{i=1}^nE[X_{nij}]=\sum_{j=1}^nE[X_{nij}]=0\) for each \(n\) and all moments exist, and let \((\pi_n)_{n=2,3,\ldots}\) be a sequence such that \(\pi_n\) is a permutation of \(\{1,2,\ldots,n\}\) chosen uniformly at random from the set of all such permutations. Letting \(S_n=\sum_{i=1}^nX_{ni\pi_n(i)}\), the author shows that under a Linnik-type condition on the \(X_{nij}\) there are sequences \((u_n)_{n=2,3,\ldots}\) such that
\[
P\left(S_n\geq u_n\sqrt{\frac{1}{n}\sum_{i,j=1}^nE[X_{nij}^2]}\right)\sim1-\Phi(u_n)\,
\]
as \(n\to\infty\), where \(\Phi\) is the distribution function of a standard normal random variable. The proof uses the truncation method.
Reviewer: Fraser Daly (Edinburgh)Limit theorems and large deviations for \(\beta\)-Jacobi ensembles at scaling temperatureshttps://zbmath.org/1526.600252024-02-15T19:53:11.284213Z"Ma, Yu Tao"https://zbmath.org/authors/?q=ai:ma.yutaoSummary: Let \(\lambda = (\lambda_1,\dots, \lambda_n)\) be \(\beta\)-Jacobi ensembles with parameters \(p_1\), \(p_2\), \(n\) and \(\beta\) while \(\beta\) varying with \(n\). Set \(\gamma = {\lim_{n \to \infty}} \frac{n}{p_1}\) and \(\sigma = {\lim_{n \to \infty}} \frac{p_1}{p_2}\). In this paper, supposing \({\lim_{n \to \infty}} \frac{\log n}{\beta n} = 0\), we prove that the empirical measures of different scaled \(\lambda\) converge weakly to a Wachter distribution, a Marchenko-Pastur law and a semicircle law corresponding to \(\sigma \gamma > 0\), \(\sigma = 0\) or \(\gamma = 0\), respectively. We also offer a full large deviation principle with speed \(\beta n^2\) and a good rate function to precise the speed of these convergences. As an application, the strong law of large numbers for the extremal eigenvalues of \(\beta\)-Jacobi ensembles is obtained.Uniform large deviation principles of fractional stochastic reaction-diffusion equations on unbounded domainshttps://zbmath.org/1526.600262024-02-15T19:53:11.284213Z"Wang, Bixiang"https://zbmath.org/authors/?q=ai:wang.bixiangSummary: This paper is concerned with uniform large deviation principles of fractional stochastic reaction-diffusion equations driven by additive noise defined on unbounded domains where the solution operator is non-compact and hence the result of \textit{M. Salins} [Probab. Surv. 16, 99--142 (2019; Zbl 1481.60068)] does not apply. The nonlinear drift is assumed to be locally Lipschitz continnous instead of being globally Lipschitz continuous. We first prove a large deviation principle for a fractional linear stochastic equation by the weak convergence method, and then show a uniform large deviation principle for the fractional nonlinear equation by a uniform contraction principle, despite the Sobolev embeddings are non-compact in unbounded domains. The result of the paper regrading the uniform large deviations can be applied to investigate the exit time and exit place of the solutions of the stochastic reaction-diffusion equations from a given domain in the phase space.A strong law of large numbers under sublinear expectationshttps://zbmath.org/1526.600272024-02-15T19:53:11.284213Z"Song, Yongsheng"https://zbmath.org/authors/?q=ai:song.yongshengSummary: We consider a sequence of independent and identically distributed (i.i.d.) random variables \(\{\xi_k\}\) under a sublinear expectation \(\mathbb{E} = \sup_{P\in\Theta}E_P\). We first give a new proof to the fact that, under each \(P\in\Theta\), any cluster point of the empirical averages \(\bar{\xi}_n = (\xi_1+\cdots+\xi_n)/n\) lies in \([\underline{\mu}, \overline{\mu}]\) with \(\underline{\mu} = -\mathbb{E}[-\xi_1]\), \(\overline{\mu} = \mathbb{E}[\xi_1]\). Next, we consider sublinear expectations on a Polish space \(\Omega\), and show that for each constant \(\mu\in[\underline{\mu}, \overline{\mu}]\), there exists a probability \(P_{\mu}\in\Theta\) such that
\[
\lim\limits_{n\rightarrow\infty}\bar{\xi}_n = \mu,\; P_{\mu}\text{-a.s.}, \tag{0.1}
\]
supposing that \(\Theta\) is weakly compact and \(\{\xi_n\}\in L^1_{\mathbb{E}}(\Omega)\). Under the same conditions, we obtain a generalization of (0.1) in the product space \(\Omega = \mathbb{R}^{\mathbb{N}}\) with \(\mu\in[\underline{\mu}, \overline{\mu}]\) replaced by \(\Pi = \pi(\xi_1, \cdots, \xi_d)\in [\underline{\mu}, \overline{\mu}] \). Here \(\pi\) is a Borel measurable function on \(\mathbb{R}^d\), \(d\in\mathbb{N}\). Finally, we characterize the triviality of the tail \(\sigma\)-algebra of the i.i.d. random variables under a sublinear expectation.On the \((p,q)\)-type strong law of large numbers for sequences of independent random variableshttps://zbmath.org/1526.600282024-02-15T19:53:11.284213Z"Thành, Lê Vǎn"https://zbmath.org/authors/?q=ai:le-van-thanh.Summary: \textit{D. Li} et al. [Trans. Am. Math. Soc. 368, No. 1, 539--561 (2016; Zbl 1335.60041)] introduced a refinement of the Marcinkiewicz-Zygmund strong law of large numbers (SLLN), the so-called \((p,q)\)-type SLLN, where \(0<p<2\) and \(q>0\). They obtained sets of necessary and sufficient conditions for this new type SLLN for two cases: \(0<p<1, q>p\), and \(1\leq p<2,q\geq 1\). Results for the case where \(0<q\leq p<1\) and \(0<q<1\leq p<2\) remain open problems. This paper gives a complete solution to these problems. We consider random variables taking values in a real separable Banach space \(\mathbf{B}\), but the results are new even when \(\mathbf{B}\) is the real line. Furthermore, the conditions for a sequence of random variables \(\left\lbrace X_n, n \geq 1\right\rbrace\) satisfying the \((p, q)\)-type SLLN are shown to provide an exact characterization of stable type \(p\) Banach spaces.
{{\copyright} 2022 Wiley-VCH GmbH.}A functional limit theorem for self-normalized linear processes with random coefficients and i.i.d. heavy-tailed innovationshttps://zbmath.org/1526.600292024-02-15T19:53:11.284213Z"Krizmanić, Danijel"https://zbmath.org/authors/?q=ai:krizmanic.danijelThe author considers a linear process of the following form: \(X_{i}=\sum_{j=0}^{\infty }C_{j}Z_{i-j}\), \(i\in\mathbb{Z}\), where \((Z_{i})\) is a strictly stationary sequence of random variables (r.v.) with regularly varying tail \(P(\left\vert Z_{i}\right\vert
>x)=x^{-\alpha }L(x)\),\(x>0\) and \(0<\alpha <2\), and \((C_{i})\) is a sequence of r.v. independent of \((Z_{i})\) and so that the above series is a.s. convergent. Let \((a_{n})\) be a sequence of positive numbers such that \(nP(\left\vert Z_{1}\right\vert >a_{n})\rightarrow 1\). In the paper, the author considers the functional convergence of the joint stochastic process \(L_{n}(t)=(a_{n}^{-1}\sum_{i=1}^{\left[ nt\right] }X_{i},a_{n}^{-2}
\sum_{i=1}^{\left[ nt\right] }X_{i}^{2})\), \(0\leq t\leq 1\) using the weak Skorohod \(M_{2}\) topology. From this result the author obtains a functional convergence result of the self-normalized sum \(\xi _{n}^{-1}\sum_{i=1}^{\left[ nt \right] }X_{i}\), where \(\xi _{n}^{2}=\sum_{i=1}^{n}X_{i}^{2}\).
Reviewer: Edward Omey (Brussel)High-dimensional regimes of non-stationary Gaussian correlated Wishart matriceshttps://zbmath.org/1526.600302024-02-15T19:53:11.284213Z"Bourguin, Solesne"https://zbmath.org/authors/?q=ai:bourguin.solesne"Dang, Thanh"https://zbmath.org/authors/?q=ai:dang.thanh-ha|dang.thanh-tung|dang.thanh-hai|dang.thanh-hungSummary: We study the high-dimensional asymptotic regimes of correlated Wishart matrices \(d^{- 1}\mathcal{Y} \mathcal{Y}^T\), where \(\mathcal{Y}\) is a \(n\times d\) Gaussian random matrix with correlated and non-stationary entries. We prove that under different normalizations, two distinct regimes emerge as both \(n\) and \(d\) grow to infinity. The first regime is the one of central convergence, where the law of the properly renormalized Wishart matrices becomes close in Wasserstein distance to that of a Gaussian orthogonal ensemble matrix. In the second regime, a non-central convergence happens, and the law of the normalized Wishart matrices becomes close in Wasserstein distance to that of the so-called Rosenblatt-Wishart matrix recently introduced by Nourdin and Zheng. We then proceed to show that the convergences stated above also hold in a functional setting, namely as weak convergence in \(C([a,b]; M_n(\mathbb{R}))\). As an application of our main result (in the central convergence regime), we show that it can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. Our findings complement and extend a rich collection of results on the study of the fluctuations of Gaussian Wishart matrices, and we provide explicit examples based on Gaussian entries given by normalized increments of a bi-fractional or a sub-fractional Brownian motion.A new integral equation for Brownian stopping problems with finite time horizonhttps://zbmath.org/1526.600312024-02-15T19:53:11.284213Z"Christensen, Sören"https://zbmath.org/authors/?q=ai:christensen.soren.1|christensen.soren-torholm|christensen.soren.2|christensen.soren-gram"Fischer, Simon"https://zbmath.org/authors/?q=ai:fischer.simonThe subject of the paper is the classical finite time horizon stopping problems driven by a Brownian motion:
\[
V(t, x) = \sup_{t\leq\tau\leq 0}{\mathbf E}_{(t,x)}[g(\tau,W_\tau )],
\]
where \(W\) is an n-dimensional standard Brownian motion started at time \(t\leq 0\) in \(x\in\mathbb{R}^n\) and \(g: \mathbb{R}_{\leq 0} \times\mathbb{R}^n\rightarrow\mathbb{R}\) a payoff function [\textit{T. L. Lai} and \textit{T. W. Lim}, J. Stat. Plann. Inference 130, No. 1--2, 21--47 (2005; Zbl 1085.62096); \textit{G. Peskir} and \textit{A. Shiryaev}, Optimal stopping and free-boundary problems. Basel: Birkhäuser (2006; Zbl 1115.60001)]. A new class of Fredholm-type integral equations for the stopping set is derived. For a large class of discounted problems, analytical arguments show that the equation uniquely characterizes the stopping boundary of the problem. Regardless of the uniqueness, the representation is used to rigorously find the limit behavior of the stopping boundary close to the terminal time. It turns out that the leading-order coefficient is universal for a wide range of problems. This discussion shows that the representation can be used for numerical purposes.
The authors are confident that uniqueness holds in a more general setting as well, but this still has to be proven.
Reviewer: Krzysztof J. Szajowski (Wrocław)Secretary problem with hidden information; searching for a high merit candidatehttps://zbmath.org/1526.600322024-02-15T19:53:11.284213Z"Kubicka, Ewa M."https://zbmath.org/authors/?q=ai:kubicka.ewa-m"Kubicki, Grzegorz"https://zbmath.org/authors/?q=ai:kubicki.grzegorz-m"Kuchta, Małgorzata"https://zbmath.org/authors/?q=ai:kuchta.malgorzata"Morayne, Michał"https://zbmath.org/authors/?q=ai:morayne.michalThe subject of the paper are consideration related to some generalization of the secretary problem [\textit{T. S. Ferguson}, Stat. Sci. 4, No. 3, 282--296 (1989; Zbl 0788.90080)].
The basic model of the secretary problem is formulated for the sequential observation of permutations of different numbers, assuming that all permutations are equally likely. Furthermore, the authors additionally assumed that these numbers are randomly selected, and for the sake of consideration, they are practically sequentially added observations of independent random variables \(X_1,X_2,\ldots,X_n\) with a uniform distribution on \([0,1]\). This allows them to formulate a selection problem with a criterion referring to the value of this sequence but limiting the decision maker's knowledge to relative ranks of them. For the topics covered in the work consistent with the literature on the subject, we will introduce the random variables \(Y_k=\textbf{card}\{1\leq i\leq k: X_i<X_k\} \) (i.e., the sequence of relative ranks for the sequentially delivered random variables \(\{X_i\}_{i=1}^n\)). It is known that \(\{Y_i\}_{i=1}^n\) are independent, having a uniform distribution: \(\textbf{P}\{Y_i=j\}= \frac{1}{i}\) for \(j=1,2,\ldots,i\), \(i=1,2,\ldots,n\). Based on stopping times with respect to filtering \({\mathcal F}_k= \sigma\{Y_1,\ldots,Y_k\}\), \(k=1,2,\ldots,n\), looking for a stopping moment that maximizes \(\textbf{P}(X_\tau<a)\), where \( a=\frac{1}{n}\). In essence, the problem was reduced to the optimal stopping of a sequence of independent random variables \(\{Y_i\}_{i=1}^n\) with the payoff function \(F_{r,k}( \frac{1}{n})=\textbf{P}(X_k\leq \frac{1}{n}|Y_k=r)\)--the value of the distribution function of the \(r\)th order statistic for the set of \(k\)-elements sample [\textit{A. Stepanov}, Commun. Math. 29, No. 1, 151--162 (2021; Zbl 1481.60099)], that is, to determine \(\textbf{P}(X_{\tau^\star}<\frac{1}{n})=\sup_\tau\textbf{E}F_{Y_\tau\tau}(\frac{1}{n })\) and the strategy \(\tau^\star\). In the paper, the problem is solved. The construction of the solution gives some combinatorial identities. Value-added examines of the asymptotics of the strategy and the value of the problem were provided.
\textsl{Reviewer's remark}: The role of information that the decision maker has in the selection problem related to the secretary problem was the subject of research by \textit{E. G. Enns} [J. Am. Stat. Assoc. 70, 640--643 (1975; Zbl 0308.62082)], where decision maker has less information than relative ranks (cf. also [\textit{W. Stadje}, Math. Methods Oper. Res. 45, No. 1, 119--131 (1997; Zbl 0880.90001); \textit{M. Sakaguchi} and \textit{K. Szajowski}, Math. Japon. 45, No. 3, 483--495 (1997; Zbl 0895.90198)]).
Reviewer: Krzysztof J. Szajowski (Wrocław)Dual spaces for martingale Musielak-Orlicz Lorentz Hardy spaceshttps://zbmath.org/1526.600332024-02-15T19:53:11.284213Z"Weisz, Ferenc"https://zbmath.org/authors/?q=ai:weisz.ferenc"Xie, Guangheng"https://zbmath.org/authors/?q=ai:xie.guangheng"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachunThis paper deals with various subspaces of martingales on a stochastic probability space with discrete time. A Musielak-Orlicz space on the underlying probability space generalizes the \(L^p\)-space by a more informative locally depending function instead of a global parameter \(p \ge 1\), and a Musielak-Orlicz Lorentz space controls further by a parameter \(q > 0\). Five martingale spaces are considered, which are called Martingale Musielak-Orlicz Lorentz Hardy spaces, depending on whether the maximal function, the quadratic variation and conditional quadratic variation, etc. of a martingale lies in the Musielak-Orlicz Lorentz space.
An atomic representation of such a martingale is one which is given by an infinite sum of special simpler functions. The authors show how, under some technical assumptions, these martingales of the martingale Musielak-Orlicz Lorentz Hardy spaces can be presented by such infinite sums of atoms. For the atomic presentations it is required that the sigma algebras of the filtration of the underlying probability space is generated by countably many atoms.
After some quasinorm estimates among the various martingale spaces, the paper continues with the introduction of martingale Musielak-Orlicz BMO-type spaces, and show how they are the dual spaces of martingale Musielak-Orlicz Lorentz Hardy spaces specified by conditional quadratic variation and maximal function, respectively.
Even if the paper appears somewhat technical the authors emphasize in a last discussion how the requirements can be easily checked in certain cases and how a vast class of similar martingale Hardy spaces known from the literature are covered by their paper.
The proof is by longer direct estimates and computations, and stopping times play an important role.
Reviewer: Bernhard Burgstaller (Florianópolis)On the law of terminal value of additive martingales in a remarkable branching stable processhttps://zbmath.org/1526.600342024-02-15T19:53:11.284213Z"Yang, Hairuo"https://zbmath.org/authors/?q=ai:yang.hairuoLet \((Z_n)_{n\ge 0}\) be a supercritical branching random walk on \(Q:=[0,\infty)\times[0,\infty)\) starting at time \(0\) with one particle at the origin of \(Q\). At time 1 this particle is replaced by a set of particles whose positions in \(Q\) are given by the atoms of a Poisson point process \(\mathcal{Z}\) with the Lebesgue measure as its intensity measure. At time \(n>1\) every particle present at time \(n-1\) is independently replaced by a set of particles whose displacements from it are given by the atoms of an independent copy of \(\mathcal{Z}\). Then
\[W_n:=\left(\mathbf{E} \int_Q e^{-(x+y)}Z_1(dx\,dy)\right)^{-n} \left(\int_Q e^{-(x+y)} Z_n(dx\,dy)\right)\]
is an additive martingale satisfying the conditions for \(L^1\)-convergence to its limit value \(W\), see [\textit{R. Lyons}, IMA Vol. Math. Appl. 84, 217--221 (1997; Zbl 0897.60086)].
The author characterizes the law of \(W\) as follows: Define \(\phi(s):=\mathbf{E}e^{sW}\) for all \(s\) such that \(\phi(s)\) is finite, \(r^*:= \exp\left(\int^\infty_0 ((2(e^u-u-1))^{-1/2}- u^{-1})\,du\right)\), and \(F(y):= \int^\infty_{\log y} (2(e^u-u-1))^{-1/2}du\), \(y>1\). Then \(r^*=F^{-1}(\log r^*-\log r)\) for all \(r\in(0,r^*)\) and \(\sup\{r\in\mathbb{R}:\phi(r)<\infty\}= r^*\).
Furthermore, the law of \(W\) is self-decomposable and admits an unimodal density function \(f\) such that \(-\log f(x)\sim r^*x\) as \(x\to\infty\) and \(-\log f(x)\sim(\log x)^2/2\) as \(x\to 0+\). Before specializing to and solving the problem in his particular setting, the author leads up to it in a more general framework of related literature.
Reviewer: Heinrich Hering (Rockenberg)Energy-constrained random walk with boundary replenishmenthttps://zbmath.org/1526.600352024-02-15T19:53:11.284213Z"Wade, Andrew"https://zbmath.org/authors/?q=ai:wade.andrew-r"Grinfeld, Michael"https://zbmath.org/authors/?q=ai:grinfeld.michaelSummary: We study an energy-constrained random walker on a length-\(N\) interval of the one-dimensional integer lattice, with boundary reflection. The walker consumes one unit of energy for every step taken in the interior, and energy is replenished up to a capacity of \(M\) on each boundary visit. We establish large \(N, M\) distributional asymptotics for the lifetime of the walker, i.e., the first time at which the walker runs out of energy while in the interior. Three phases are exhibited. When \(M \ll N^2\) (energy is scarce), we show that there is an \(M\)-scale limit distribution related to a Darling-Mandelbrot law, while when \(M \gg N^2\) (energy is plentiful) we show that there is an exponential limit distribution on a stretched-exponential scale. In the critical case where \(M / N^2 \rightarrow \rho \in (0,\infty)\), we show that there is an \(M\)-scale limit in terms of an infinitely-divisible distribution expressed via certain theta functions.Max-compound Cox processes. IIIhttps://zbmath.org/1526.600362024-02-15T19:53:11.284213Z"Korolev, V. Yu."https://zbmath.org/authors/?q=ai:korolev.victor-yu"Sokolov, I. A."https://zbmath.org/authors/?q=ai:sokolov.i-a"Gorshenin, A. K."https://zbmath.org/authors/?q=ai:gorshenin.a-kSummary: Extreme values are considered in samples with random size that have a mixed Poisson distribution being generated by a doubly stochastic Poisson process. We prove some inequalities providing bounds on the rate of convergence in limit theorems for the distributions of max-compound Cox processes.
For Part II see [the authors, J. Math. Sci. 246, No. 4, 488--502 (2020; \url{doi:0.1007/s10958-020-04754-9})]. For Part I see [the authors, J. Math. Sci., New York 237, No. 6, 789--803 (2019; Zbl 1478.60149)].Exponential discrete gradient schemes for a class of stochastic differential equationshttps://zbmath.org/1526.600372024-02-15T19:53:11.284213Z"Ruan, Jialin"https://zbmath.org/authors/?q=ai:ruan.jialin"Wang, Lijin"https://zbmath.org/authors/?q=ai:wang.lijin"Wang, Pengjun"https://zbmath.org/authors/?q=ai:wang.pengjunA class of exponential discrete gradient schemes for SDEs with linear and gradient components in the coefficients is proposed. The authors investigate their root mean-square errors and some structure-preserving properties of those schemes for SDEs. Some numerical tests and illustrations supporting their findings are given too.
Reviewer: Henri Schurz (Carbondale)The compact support property for solutions to the stochastic partial differential equations with colored noisehttps://zbmath.org/1526.600382024-02-15T19:53:11.284213Z"Han, Beom-Seok"https://zbmath.org/authors/?q=ai:han.beom-seok"Kim, Kunwoo"https://zbmath.org/authors/?q=ai:kim.kunwoo"Yi, Jaeyun"https://zbmath.org/authors/?q=ai:yi.jaeyunSummary: We study the compact support property for nonnegative solutions of the following stochastic partial differential equations (SPDEs): \(\partial_t u=a^{ij}u_{x^ix^j}(t,x)+b^iu_{x^i}(t,x)+cu+h(t,x,u(t,x))\dot{F}(t,x), (t,x)\in (0,\infty) \times \mathbb{R}^d\), where \(\dot{F}\) is a spatially homogeneous Gaussian noise that is white in time and colored in space, and \(h(t, x, u)\) satisfies \(K^{-1}u^\lambda \leq h(t, x, u) \leq K(1+u)\) for \(\lambda \in (0,1)\) and \(K\geq 1\). We show that if the initial data \(u_0\geq 0\) has a compact support, then, under the \textit{reinforced} Dalang's condition on \(\dot F\) (which guarantees the existence and the Hölder continuity of a weak solution), all nonnegative weak solutions \(u(t, \cdot)\) have the compact support for all \(t> 0\) with probability 1. Our results extend the works by \textit{C. Mueller} and \textit{E. A. Perkins} [Probab. Theory Relat. Fields 93, No. 3, 325--358 (1992; Zbl 0767.60054)] and \textit{N. V. Krylov} [Probab. Theory Relat. Fields 108, No. 4, 543--557 (1997; Zbl 0886.60061)], in which they show the compact support property only for the one-dimensional SPDEs driven by space-time white noise on \((0, \infty)\times \mathbb{R}\).Dynamics of a stochastic fractional nonlocal reaction-diffusion model driven by additive noisehttps://zbmath.org/1526.600392024-02-15T19:53:11.284213Z"Li, Lingyu"https://zbmath.org/authors/?q=ai:li.lingyu"Chen, Zhang"https://zbmath.org/authors/?q=ai:chen.zhang"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomasSummary: In this paper, we are concerned with the long-time behavior of stochastic fractional nonlocal reaction-diffusion equations driven by additive noise. We use the techniques of random dynamical systems to transform the stochastic model into a random one. To deal with the new nonlocal term appeared in the transformed equation, we first use a generalization of Peano's theorem to prove the existence of local solutions, and then adopt the Galerkin method to prove existence and uniqueness of weak solutions. Next, the existence of pullback attractors for the equation and its associated Wong-Zakai approximation equation driven by colored noise are shown, respectively. Furthermore, we establish the upper semi-continuity of random attractors of the Wong-Zakai approximation equation as \(\delta\rightarrow0^+ \).Stochastic equations and equations for probabilistic characteristics of processes with damped jumpshttps://zbmath.org/1526.600402024-02-15T19:53:11.284213Z"Mel'nikova, Irina Valer'yanovna"https://zbmath.org/authors/?q=ai:melnikova.irina-valeryanovna"Bovkun, Vadim Andreevich"https://zbmath.org/authors/?q=ai:bovkun.vadim-andreevichSummary: Processes such as shot noise are an adequate tool for modeling discontinuous random processes with a damping effect. Such processes arise in various fields of physics, technology and human activity, including the field of finance. They allow to simulate not only abrupt changes in the values of processes with jumps, but also the subsequent return of values to their original or near the original position. For this reason, shot noise is a suitable tool for simulating price hikes of various market assets. This paper presents a general formulation of the shot noise type processes, a stochastic differential equation corresponding to the process, and a connection with the integral-differential equation for the transition probability density, which is formulated in terms of generalized functions. In addition, the paper considers a stochastic equation that allows, along with an abrupt change described by a process of the shot noise type, a continuous change in the process. An equation is obtained for the transition probability density of a process containing jump-like and continuous types of evolution, which should also be considered in spaces of generalized functions. Thus, main results of the paper are obtained on the basis of the application of stochastic analysis and the theory of generalized functions.On the stochastic evolution equation driven by Brownian motion in a separable spacehttps://zbmath.org/1526.600412024-02-15T19:53:11.284213Z"Miraoui, Mohsen"https://zbmath.org/authors/?q=ai:miraoui.mohsen"Missaoui, Sonia"https://zbmath.org/authors/?q=ai:missaoui.soniaSummary: This article discusses issues surrounding the concept of a double measure Stepanov-type pseudo-almost periodic (`pap') mean-square process with a double measures. Moreover, using stochastic analysis techniques and Banach's fixed point Theorem, we investigate the uniqueness and existence of the `pap' solution to partial stochastic neutral differential equations with double measure mean-square 'pap' coefficients of the Stepanov type driven by the Brownian motion in a separable Hilbert space \(\mathcal{K}\). Therefore, we study its global exponential stability. The concluding segment of our work is exemplified by a practical illustration, affirming the reliability and applicability of our findings.Convergence of Gibbs sampling: coordinate hit-and-run mixes fasthttps://zbmath.org/1526.600422024-02-15T19:53:11.284213Z"Laddha, Aditi"https://zbmath.org/authors/?q=ai:laddha.aditi"Vempala, Santosh S."https://zbmath.org/authors/?q=ai:vempala.santosh-sSummary: Gibbs sampling, also known as Coordinate Hit-and-Run (CHAR), is a Markov chain Monte Carlo algorithm for sampling from high-dimensional distributions. In each step, the algorithm selects a random coordinate and re-samples that coordinate from the distribution induced by fixing all the other coordinates. While this algorithm has become widely used over the past half-century, guarantees of efficient convergence have been elusive. We show that the Coordinate Hit-and-Run algorithm for sampling from a convex body \(K\) in \({\mathbb{R}}^n\) mixes in \(O^*(n^9 R^2/r^2)\) steps, where \(K\) contains a ball of radius \(r\) and \(R\) is the average distance of a point of \(K\) from its centroid. We also give an upper bound on the conductance of Coordinate Hit-and-Run, showing that it is strictly worse than Hit-and-Run or the Ball Walk in the worst case.Krause mean processes generated by cubic stochastic diagonally primitive matriceshttps://zbmath.org/1526.600432024-02-15T19:53:11.284213Z"Saburov, Khikmat"https://zbmath.org/authors/?q=ai:saburov.khikmat-khazhibaevichSummary: A multi-agent system is a system of multiple interacting entities, known as \textit{intelligent agents}, who possibly have different information and/or diverging interests. The agents could be robots, humans, or human teams. Opinion dynamics is a process of individual opinions in which a group of interacting agents continuously fuse their opinions on the same issue based on established rules to reach a \textit{consensus} at the final stage. Historically, the idea of reaching consensus through repeated averaging was introduced by DeGroot for a structured time-invariant and synchronous environment. Since then, consensus, which is the most ubiquitous phenomenon of multi-agent systems, has become popular in various scientific fields such as biology, physics, control engineering, and social science. To some extent, a \textit{Krause mean process} is a general model of opinion sharing dynamics in which the opinions are represented by vectors. In this paper, we represent opinion sharing dynamics by means of \textit{Krause mean processes} generated by \textit{diagonally primitive cubic doubly stochastic matrices}, and then we establish a consensus in the multi-agent system.Continuous-time stochastic analysis of rumor spreading with multiple operationshttps://zbmath.org/1526.600442024-02-15T19:53:11.284213Z"Castella, François"https://zbmath.org/authors/?q=ai:castella.francois"Sericola, Bruno"https://zbmath.org/authors/?q=ai:sericola.bruno"Anceaume, Emmanuelle"https://zbmath.org/authors/?q=ai:anceaume.emmanuelle"Mocquard, Yves"https://zbmath.org/authors/?q=ai:mocquard.yvesSummary: In this paper, we analyze a new asynchronous rumor spreading protocol to deliver a rumor to all the nodes of a large-scale distributed network. This protocol relies on successive pull operations involving \(k\) different nodes, with \(k \geq 2\), and called \(k\)-pull operations. Specifically during a \(k\)-pull operation, an uninformed node \(a\) contacts \(k-1\) other nodes at random in the network, and if at least one of them knows the rumor, then node \(a\) learns it. We perform a detailed study in continuous-time of the total time \(\Theta_{k, n}\) needed for all the \(n\) nodes to learn the rumor. These results extend those obtained in a previous paper which dealt with the discrete-time case. We obtain the mean value, the variance and the distribution of \(\Theta_{k, n}\) together with their asymptotic behavior when the number of nodes \(n\) tends to infinity.Rate of convergence of the perturbed diffusion process to its unperturbed limithttps://zbmath.org/1526.600452024-02-15T19:53:11.284213Z"Cuong, Tran Manh"https://zbmath.org/authors/?q=ai:cuong.tran-manh"Dung, Nguyen Tien"https://zbmath.org/authors/?q=ai:dung.nguyen-tienThe authors consider a variant of Brownian motion, defined by
\[
X_t= x_0 +\int_0^t b(X_s) ds +\int_0^t \sigma(X_s) d B_s+ \alpha \max_{0 \leq s \leq t} X_s,
\]
for some perturbation parameter \(\alpha<1\), where \(B_s\) is the standard Brownian motion and where \(b\) and \(\sigma\) are Lipschitz continuous functions.
This stochastic differential equation has a unique solution. Let \(Z_t\) be its solution for \(\alpha=0\), that is, \(Z_t\) is the unperturbed diffusion process associated to the initial perturbed diffusion process \(X_t\).
The authors prove that
\[
\mathbb{E} \left[ \max_t |X_t-Z_t|^p \right] \leq C |\alpha|^p.
\]
What is more, under some Hölder-like conditions (with exponent \(\beta\)) on the fonctions \(b'\) and \(\sigma'\), they also get that
\[
\mathbb{E}\left[ \max_t \left|\frac{X_t-Z_t}{\alpha}-Y_t\right|^p \right] \leq C (|\alpha|^p+|\alpha|^{p\beta}).
\]
They also give a weak convergence result: For any continuous and bounded function \(g\), one has
\[
\mathbb{E}[g(X_t)]-\mathbb{E}[g(Z_t)] \leq C \times |\alpha| \times \|g\|_\infty \times f(t),
\]
where \(f\) is a function detailed in the article. They also get an expression for the limit \(\lim_{\alpha\rightarrow0} \frac{\mathbb{E}[g(X_t)]-\mathbb{E}[g(Z_t)]}{\alpha} \) in terms of a quantity involving a Malliavin derivative and a Skorohod integral.
The proofs involve Gronwall's lemma, Hölder and Burkhodder-Davis-Gundy inequalities, and Malliavin calculus.
Reviewer: Cyril Banderier (Paris)Reflecting Brownian motion and the Gauss-Bonnet-Chern theoremhttps://zbmath.org/1526.600462024-02-15T19:53:11.284213Z"Du, Weitao"https://zbmath.org/authors/?q=ai:du.weitao"Hsu, Elton P."https://zbmath.org/authors/?q=ai:hsu.elton-pThis paper treats the well-known Gauss-Bonnet-Chern theorem for a compact manifold with boundary. The authors adopt a probabilistic approach to use reflecting Brownian motion (RBM) in order to prove the theorem. One of the peculiar features consists in the fact that the boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary local time of RBM for small times.
More precisely, let \(M\) be a smooth oriented manifold with boundary, and let \(\chi (M)\) be its Euler characteristic defined by
\[
\chi (M) = \sum_{i=0}^m (-1)^i b_i,
\]
where \(b_i = \dim H^i(M)\) are the Betti numbers, and \(H^i(M)\) is the \(i\)-dimensional cohomology group. Assume that the manifold \(M\) is equipped with a Riemannian metric. A fundamental result in differential geometry is the Gauss-Bonnet-Chern (GBC) theorem, cf. [\textit{S.-S. Chern}, Ann. Math. (2) 45, 747--752 (1944; Zbl 0060.38103)] for a simple proof of the GBC theorem for a closed Riemannian manifolds. The theorem expresses the Euler characteristic as the sum of two integrals on \(M\) and on its boundary \(\partial M\)
\[
\chi(M) = \int_M e_M(x) dx + \int_{ \partial M} e_{\partial M} ( \bar{x} ) d \bar{x},
\]
where \(e_M\) (resp. \(e_{ \partial M}\) ) is a local geometric invariant determined by the curvature tensor (resp. the second fundamental form of the boundary of the manifold) respectively. For a manifold without boundary, \textit{H. P. McKean jun.} and \textit{I. M. Singer} [J. Differ. Geom. 1, 43--69 (1967; Zbl 0198.44301)] expressed the integral of the Euler form on the manifold in terms of the supertrace \(str\) of the heat kernal \(p^*\) of the Hodge-de Rham Laplacian on the bundle of differential forms:
\[
\chi(M) = \int_M str \,\,\, p^* (t,x,x ) dx.
\]
On the other hand, the probabilistic approach to analytic index theorems was initiated by \textit{J.-M. Bismut} [in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 491--504 (1987; Zbl 0693.58023)] for Riemannian manifolds without boundary. Roughly speaking, the above approach is based upon the fact that the heat kernel on differential forms associated with the Hodge-de Rham Laplacian of the horizontal Brownian motion on the frame bundle of the Riemannian manifold. However, Bismut's approach cannot be easily carried over to a manifold with boundary. The major difficulty lies in the fact that the multiplicative functional is discontinuous at the boundary. Another difficulty is the presence of the boundary local time, with the result that the algebraic ``fantastic cancellation (FC)'' does not occur at the path level, although FC works well in the Bismut's work. Under these circumstances, using Malliavin calculus, \textit{I. Shigekawa} et al. [Osaka J. Math. 26, No. 4, 897--930 (1989; Zbl 0704.58056)] successfully handled the singular terms on the boundary, thus giving a probabilistic proof of the Gauss-Bonnet-Chern formula. Malliavin calculus allows the heat kernel to be regarded as a generalized functional on Brownian motion which can then be expanded as an asymptotic power series in the small time parameter in a generalized sense. This probabilistic approach cannot be regarded as a direct extension of the Bismut approach because of the use of Malliavin calculus.
The purpose of the present work is to give an elementary probabilistic proof of the Gauss-Bonnet-Chern theorem without using Malliavin calculus. Their approach is heavily indebted to the previous works by \textit{E. P. Hsu} [Stochastic analysis on manifolds. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 0994.58019)] and Shigekawa et al. [loc. cit.].
Reviewer: Isamu Dôku (Saitama)On systems of particles in singular repulsive interaction in dimension one: log and Riesz gashttps://zbmath.org/1526.600472024-02-15T19:53:11.284213Z"Guillin, Arnaud"https://zbmath.org/authors/?q=ai:guillin.arnaud"Le Bris, Pierre"https://zbmath.org/authors/?q=ai:le-bris.pierre"Monmarché, Pierre"https://zbmath.org/authors/?q=ai:monmarche.pierreSummary: In this article, we prove the first quantitative uniform in time propagation of chaos for a class of systems of particles in singular repulsive interaction in dimension one that contains the Dyson Brownian motion. We start by establishing existence and uniqueness for the Riesz gases, before proving propagation of chaos with an original approach to the problem, namely coupling with a Cauchy sequence type argument. We also give a general argument to turn a result of weak propagation of chaos into a strong and uniform in time result using the long time behavior and some bounds on moments, in particular enabling us to get a uniform in time version of the result of \textit{E. Cépa} and \textit{D. Lépingle} [Probab. Theory Relat. Fields 107, No. 4, 429--449 (1997; Zbl 0883.60089)].Strong Feller and ergodic properties of the \((1+1)\)-affine processhttps://zbmath.org/1526.600482024-02-15T19:53:11.284213Z"Chen, Shukai"https://zbmath.org/authors/?q=ai:chen.shukai"Li, Zenghu"https://zbmath.org/authors/?q=ai:li.zenghuThis paper treats the strong Feller and ergodic properties of the total variation distance of the \((1+1)\)-affine process, which are derived from some estimates for the variations of transition probabilities of the process. The key strategy consists in the pathwise construction and analysis of several Markov couplings making use of strong solutions of certain type stochastic equations.
More precisely, let \(m \geq 0\) and \(n \geq 0\) be integers. A time-homogeneous \(( m + n )\)-dimensional Markov processes
\[
\{ X_t : t \geq 0 \} = \{ ( Y_t, Z_t) : t \geq 0 \}
\]
taking values in \(D = {\mathbb R}_+^m \times {\mathbb R}^n\) is called an affine Markov process if its characteristic function satisfies
\[
{\mathbb E} [ e^{ i \langle X_t, u \rangle } \vert X_0 = x ] = \exp \{ \langle x, \psi(t, i u) \rangle + \phi (t, i u) \}, \quad x \in D, \quad u \in {\mathbb R}^{m+n},
\]
where \(\phi\) and \(\psi\) satisfy certain generalized Riccati differential equations. The affine property means roughly that the logarithm of the characteristic function is affine with respect to the initial state. In this case, it is known that the \(m\)-dimensional process \(\{ Y_t : t \geq 0 \}\) is a continuous-state branching process with immigration (CBI process). Then the above formulation includes as special cases both the CBI process and the Ornstain-Uhlenbeck (OU)-type process.
In this paper the authors prove some estimates for the variations of transition probabilities of the \((1 + 1)\)-affine process. From those estimates they deduce the strong Feller and ergodic properties in the total variation distance of the process. The key strategy is to construct several Markov couplings using strong solutions of stochastic equations, which naturally extend those of the CBI process and the OU-type process introduced by \textit{Z. Li} and \textit{C. Ma} [Stochastic Processes Appl. 125, No. 8, 3196--3233 (2015; Zbl 1362.62047)]. For simplicity, here the authors only discuss the \(( 1 + 1)\)-dimensional process. The method can be modified to treat general finite-dimensional affine processes by some extra work, which are addressed separately. The Feller property implies that \(\{ X_t : t \geq 0 \}\) has a càdlàg realization. Suppose that \(\{ X_t : t \geq 0 \} = \{ ( Y_t, Z_t): t \geq 0 \}\) is a \((1+1)\)-affine process with transition semigroup \((P_t)_{ t \geq 0 }\) defined by
\[
\int_D e^{ \langle u, \nu \rangle} P_t(x, d \nu) = \exp \{ \langle x, \psi(t, u) \rangle + \phi(t,u) \},
\]
where \((\phi, \psi)\) is the unique solution of certain Riccati differential equations, with \(D = {\mathbb R}_+ \times {\mathbb R}\), \(u \in U = {\mathbb C}_- \times i {\mathbb R}\), \({\mathbb C}_- = \{ a + i b : a \in {\mathbb R}_-, \, b \in {\mathbb R} \}\), and \(i {\mathbb R} = \{ i a : a \in {\mathbb R} \}\). Then \(\{ Y_t : t \geq 0 \}\) is a Markov process on \({\mathbb R}_+\) with Feller transition semigroup \(( P_t^{(1)} )_{ t \geq 0 }\) defined by
\[
\int_{ {\mathbb R}_+ } e^{ - \lambda \nu } P_t^{(1)} (x, d \nu ) = \exp \left\{ x \psi_1(t, - \lambda, 0) + \int_0^t F( \psi_1(t, - \lambda, 0), 0 ) ds \right\}, \quad \lambda \geq 0.
\]
It is known that \(\{ Y_t : t \geq 0 \}\) is a continuous-state branching process with immigration (CBI process) with branching mechanism \(\lambda \mapsto - R(- \lambda, 0)\) and immigration mechanism \(\lambda \mapsto - F(- \lambda, 0)\). The authors derive a more accurate estimate for general finite-dimensional affine processes.
Proposition A.
Let \(\{ ( Y_t, Z_t): t \geq 0 \}\) be a \((1 + 1)\)-affine process with \(Y_0 = y \in {\mathbb R}_+\) and \(Z_0 = z \in {\mathbb R}\). Let \(D_1 = {\mathbb R}_+ \times [-1, 1]\) and \(D_1^c = D \setminus D_1\). Then
\begin{multline*}
{\mathbb E} \vert Z_t \Vert \leqslant e^{ \beta_{22} t} \vert z \vert
+ \left( \vert b_2 \vert + 2 \int_{ D_1^c} \vert \nu_2 \vert \nu (d \nu ) \right) \int_0^t e^{ \beta_{22} (t-s)} ds \\
+ \left( \vert \beta_{21} \vert + 2 \int_{ D_1^c} \vert \nu_2 \vert \mu ( d \nu) \right) \int_0^t e^{ \beta_{22} (t-s)} {\mathbb E}( Y_s ) ds
+ \left[ \sqrt{2} \sigma_0 + \left( \int_{ D_1} \nu_2^2 \cdot \nu ( d \nu ) \right)^{1/2} \right] \left( \int_0^t e^{ 2 \beta_{22} (t-s)} ds \right)^{1/2} \\
+ \sqrt{2} ( \sigma_{21} + \sigma_{22} ) \left( \int_0^te^{ 2 \beta_{22} (t-s)} {\mathbb E} ( Y_s) ds \right)^{1/2}
+ \left( \int_{D_1} \nu_2^2 \mu( d \nu ) \right)^{1/2} \left( \int_0^t e^{ 2 \beta_{22} (t-s)} {\mathbb E}( Y_s ) ds \right)^{1/2},
\end{multline*}
where \(\mu( d \nu ) = \mu ( d \nu_1, d \nu_2)\) and \(\nu( d \nu ) = \nu ( d \nu_1, d \nu_2)\) are \(\sigma\)-finite measure on \(D\), supported on \(D \setminus \{ 0 \}\), such that
\[
\int_D ( \nu_1 \wedge \nu_1^2 + \vert \nu_2 \vert \wedge \vert \nu_2 \vert^2 ) \mu ( d \nu) < \infty
\]
and
\[
\int_D ( \nu_1 + \vert \nu_2 \vert \wedge \vert \nu_2 \vert^2 ) \nu ( d \nu ) < \infty.
\]
The key estimates for the variations of the transition probabilities are established, whereby the strong Feller property and the exponential ergodicity are also deduced. Actually, it is proved that \(( P_t)_{ t \geq 0 }\) is a sgtrong Feller transition semigroup. For the unique stationary distribution for \((P_t)_{t \geq 0}\), for every \(\delta > 0\), there is a constant \(C_{\delta} \geq 0\) such that
\[
\Vert P_t (x, \cdot ) - \pi \Vert_{var} \leqslant C_{\delta} ( 1 + \vert x \vert ) e^{- kt/2},
\]
for \(t \geq \delta\), \(x \in D\), and \(k = \vert \beta_{11} \vert \wedge \vert \beta_{22} \vert\), where \(\Vert \cdot \Vert_{var}\) denotes the total variation norm. Lastly, under a weaker condition an ergodic result is proved as well.
For other related works, see, e.g., \textit{Z. Li} [Theory Probab. Appl. 66, No. 2, 276--298 (2021; Zbl 1470.60230)] for ergodicities and exponential ergodicities of Dawson-Watanabe-type processes, and \textit{E. Mayerhofer} et al. [Stochastic Processes Appl. 130, No. 7, 4141--4173 (2020; Zbl 1434.60195)] for geometric ergodicity of affine processes on cones.
Reviewer: Isamu Dôku (Saitama)QBD processes associated with Jacobi-Koornwinder bivariate polynomials and urn modelshttps://zbmath.org/1526.600492024-02-15T19:53:11.284213Z"Fernández, Lidia"https://zbmath.org/authors/?q=ai:fernandez.lidia"de la Iglesia, Manuel D."https://zbmath.org/authors/?q=ai:dominguez-de-la-iglesia.manuelSummary: We study a family of quasi-birth-and-death (QBD) processes associated with the so-called first family of Jacobi-Koornwinder bivariate polynomials. These polynomials are orthogonal on a bounded region typically known as the swallow tail. We will explicitly compute the coefficients of the three-term recurrence relations generated by these QBD polynomials and study the conditions under we can produce families of discrete-time QBD processes. Finally, we show an urn model associated with one special case of these QBD processes.The effect of loss preference on queueing with information disclosure policyhttps://zbmath.org/1526.600502024-02-15T19:53:11.284213Z"Cao, Jian"https://zbmath.org/authors/?q=ai:cao.jian"Guo, Yongjiang"https://zbmath.org/authors/?q=ai:guo.yongjiang"Hu, Zhongxin"https://zbmath.org/authors/?q=ai:hu.zhongxinAuthors' abstract: In this paper, we incorporate loss preference into an M/M/1 queueing with a threshold disclosure policy and analyse its impact on the customers' queueing strategies and the queueing system's idle stationary probability. In the queueing system, customers are strategic and divided into two groups: the informed and the uninformed. Informed customers are assumed to be fully rational, whereas uninformed customers are assumed to have loss preference. Uninformed customers with loss preference are categorized into two types according to their asymmetry perceptions, which anchor on the difference between gain and loss: loss neutrality and loss aversion. We firstly determine customers' equilibrium decisions, and then derive the idle stationary probability at equilibrium. We find that loss preference reduces the customers' joining probability, and results in a higher idle stationary probability. Furthermore, we find that for the uninformed customers with stronger loss aversion, the system manager should lower the threshold of disclosure to maintain a stable demand of uninformed customers. In addition, in the case of mixed-strategy at equilibrium, with the increase of the threshold of disclosure, the idle stationary probability increases for an underloaded queue. However, for an overloaded queue, the idle stationary probability decreases with increasing the threshold of disclosure.
Reviewer: Vyacheslav Abramov (Melbourne)The time constant for Bernoulli percolation is Lipschitz continuous strictly above \(p_c\)https://zbmath.org/1526.600512024-02-15T19:53:11.284213Z"Cerf, Raphaël"https://zbmath.org/authors/?q=ai:cerf.raphael"Dembin, Barbara"https://zbmath.org/authors/?q=ai:dembin.barbaraThe authors consider a general extension of the model of first passage percolation first introduced by \textit{J. M. Hammersley} and \textit{D. J. A. Welsh} [in: Bernoulli-Bayes-Laplace, Anniversary Vol., Proc. Int. Res. Semin., Berkeley 1963, 61--110 (1965; Zbl 0143.40402)] as a model for the spread of a fluid in a porous medium. They consider the standard model of i.i.d. first passage percolation on \(\mathbb Z^d\) with given distribution \(G\) on \([0, +\infty]\), the later being non excluded. When the mass of finite values is lower bounded by the standard critical percolation threshold in dimension \(d\), that is \(G([0, +\infty)) > p_c(d)\), it was known that a time constant \(\mu_G\) controlling shape and speed of the convergence of an invasion process does exist. The authors focuses here on the regularity properties of the map associated a graph \(G\) to its time constant \(\mu_G\). They first study the specific case of distributions of the form \(G_p = p \delta_1 + (1-p) \delta_\infty\) for \(p > p_c(d)\), for which the travel time between two points is equal to the length of the shortest path between the two points in a bond percolation of parameter \(p\). By the use of refined renormalization techniques to treat asymptotic behaviors and to get rid of some logarithmitic corrections, they extend and complete previous results from one of the authors (among others) and establish that the time constant is Lipschitz continuous as a function of \(p\) on every interval \([p_0, 1]\), where \(p_0 > p_c(d)\).
Reviewer: Arnaud Le Ny (Paris)The effect of disorder on quenched and averaged large deviations for random walks in random environments: boundary behaviorhttps://zbmath.org/1526.600522024-02-15T19:53:11.284213Z"Bazaes, Rodrigo"https://zbmath.org/authors/?q=ai:bazaes.rodrigo"Mukherjee, Chiranjib"https://zbmath.org/authors/?q=ai:mukherjee.chiranjib"Ramírez, Alejandro F."https://zbmath.org/authors/?q=ai:ramirez.alejandro-f"Saglietti, Santiago"https://zbmath.org/authors/?q=ai:saglietti.santiagoIn this paper the authors study a random walk in a uniformly elliptic and i.i.d. environment on \(\mathbb Z^d\), where \(d\geq 4\). Assuming that the disorder of the environment is low enough, it is shown that the quenched and annealed large deviation rate functions agree on any compact set contained in the boundary of their domain which does not intersect any of the \((d - 2)\)-dimensional facets of the boundary. A simple explicit formula is obtained for both rate functions on any such compact set of boundary at low enough disorder. Moreover, for a general parametrized family of environments, it is shown that the strength of disorder determines a phase transition in the equality of both rate functions, in the sense that the two rate functions agree at each point of boundary when the disorder is smaller than a specific value and disagree when it is larger.
Reviewer: Utkir A. Rozikov (Tashkent)Strong stochastic Runge-Kutta-Munthe-Kaas methods for nonlinear Itô SDEs on manifoldshttps://zbmath.org/1526.650012024-02-15T19:53:11.284213Z"Muniz, Michelle"https://zbmath.org/authors/?q=ai:muniz.michelle"Ehrhardt, Matthias"https://zbmath.org/authors/?q=ai:ehrhardt.matthias"Günther, Michael"https://zbmath.org/authors/?q=ai:gunther.michael"Winkler, Renate"https://zbmath.org/authors/?q=ai:winkler.renateSummary: In this paper we present numerical methods for the approximation of nonlinear Itô stochastic differential equations on manifolds. For this purpose, we extend Runge-Kutta-Munthe-Kaas (RKMK) schemes for ordinary differential equations on manifolds to the stochastic case and analyse the strong convergence of these schemes. Since these schemes are based on the application of a stochastic Runge-Kutta (SRK) scheme in a corresponding Lie algebra, we address the question under which circumstances the stochastic RKMK method has the same strong order of convergence as the applied SRK scheme. To illustrate our answer to this question and the effectiveness of our schemes, we show some numerical results of applying these methods to a problem with an autonomous underwater vehicle.Numerical attractors for rough differential equationshttps://zbmath.org/1526.650342024-02-15T19:53:11.284213Z"Duc, Luu Hoang"https://zbmath.org/authors/?q=ai:duc.luu-hoang"Kloeden, Peter"https://zbmath.org/authors/?q=ai:kloeden.peter-erisSummary: We study the explicit Euler scheme to approximate the solutions of rough differential equations under a bounded or linear diffusion term, where the drift term satisfies a local Lipschitz continuity and a one-sided linear growth condition. The Euler scheme is then proved to converge for a given solution, where the convergence rate is independent of the initial condition. For a dissipative drift term with a linear growth condition and a bounded diffusion term, the numerical solution under a regular grid generates a random dynamical system (RDS) which admits a random pullback attractor. We also prove that for bounded drift and diffusion terms and under a centered Gaussian noise with stationary increments, the numerical pullback attractor then converges upper semicontinuously to the continuous-time pullback attractor as the time step goes to zero.Exponential integrator for stochastic strongly damped wave equation based on the Wong-Zakai approximationhttps://zbmath.org/1526.650482024-02-15T19:53:11.284213Z"Wang, Yibo"https://zbmath.org/authors/?q=ai:wang.yibo"Cao, Wanrong"https://zbmath.org/authors/?q=ai:cao.wanrongSummary: We consider strong convergence of numerical approximations for a stochastic strongly damped wave equation driven by a class of additive space-time noises characterized by a parameter \(\beta\in(-1,2]\). With the help of the Wong-Zakai (WZ) approximation to the noise, i.e., replacing the driven noise with its finite spectral expansion truncation, we obtain an approximate equation. Based on the consistency and high regularity of the approximate equation, we develop two exponential integrators for time-stepping discretization and use the spectral Galerkin method in space to develop full-discrete schemes. Performing error estimates in the strong sense, we show that the optimal strong order in time of the proposed WZ-approximation-based exponential Euler scheme is \(\min\{1,1+\beta/2-\varepsilon\}\), which is \(|\beta|/2\) order higher than the order of \(\min\{1,1+\beta\}\) in the existing works when \(\beta\in(-1,0)\). Moreover, we prove that the proposed WZ-approximation-based exponential trapezoidal scheme is of \(\min\{3/2,1+\beta/2-\varepsilon\}\)-order strong convergence in time when \(\beta\in[-1/4,2]\), so it can break the first order barrier and obtain a higher accurate numerical solution. Numerical examples are performed to verify and develop the theoretical findings.Mixing time and simulated annealing for the stochastic cellular automatahttps://zbmath.org/1526.680072024-02-15T19:53:11.284213Z"Fukushima-Kimura, Bruno Hideki"https://zbmath.org/authors/?q=ai:fukushima-kimura.bruno-hideki"Handa, Satoshi"https://zbmath.org/authors/?q=ai:handa.satoshi"Kamakura, Katsuhiro"https://zbmath.org/authors/?q=ai:kamakura.katsuhiro"Kamijima, Yoshinori"https://zbmath.org/authors/?q=ai:kamijima.yoshinori"Kawamura, Kazushi"https://zbmath.org/authors/?q=ai:kawamura.kazushi"Sakai, Akira"https://zbmath.org/authors/?q=ai:sakai.akiraThere are several occasions in real life when we have to choose quickly one in a certain sense optimal option among extremely many options. The so-called combinatorial optimization problems are ubiquitous and possibly quite hard to be solved in a fast way. A possible approach as an attempt to provide an optimal solution for a given problem would be to translate it into the problem of minimizing the Hamiltonian of an Ising model on a finite graph \(G = (V, E)\) such that each of its ground states corresponds to an optimal solution to the original problem and vice versa. The standard approach for this kind of problem is the application of algorithms that rely on single-spin-flip Markov chain Monte Carlo methods, such as the simulated annealing based on Glauber or Metropolis dynamics.
The motivation for the main topic of this paper is a method that is actually widely used in several real-world applications -- the simulated annealing algorithm. The authors investigate a particular class of probabilistic (or stochastic) cellular automata (SCA), in which all spins are updated independently and simultaneously. It is proved that (i) if the temperature is fixed sufficiently high, then the mixing time is at most of order \(\log |V|\), and that (ii) if the temperature drops in time \(t\) as \(1/\log t\), then the limiting measure is uniformly distributed over the ground states. The former result implies faster mixing than conventional single-spin-flip MCs, such as the Glauber dynamics, while the latter result refers to the applicability of the standard temperature-cooling schedule in the simulated annealing. Although both results are proven mathematically rigorously, the authors provide reasons why they may be difficult to be directly applied in practice.
The paper is well structured into three main parts: 1, Mixing time for the SCA: In this section it is shown that the mixing in the SCA is faster than in the Glauber dynamics when the temperature is sufficiently high. 2, Simulated annealing for the SCA: In this section it is shown that if the inverse temperature increases in time \(t\) as \(\propto\log t\), then the simulated annealing for the SCA weakly converges to the uniform distribution over the set of the ground states at which the Hamiltonian attains its minimum value. 3, Comparisons and simulations: In practice, to avoid applying logarithmic cooling schedules and having to wait for long execution times, faster cooling is often considered. In this section, a variant of the SCA called \(\varepsilon\)-SCA is introduced and a quick comparison regarding effectiveness in obtaining the ground states is made. In a series of examples it is shown that simulated annealing with exponential cooling schedules can be successfully applied to the Glauber dynamics, SCA, and \(\varepsilon\)-SCA. The results are consistent with the preliminary findings established by the same authors, which reveal that the SCA outperforms Glauber dynamics in most of the scenarios, while \(\varepsilon\)-SCA outperforms both algorithms in all tested models.
Reviewer: Ctirad Matonoha (Praha)Complementarity in quantum walkshttps://zbmath.org/1526.810012024-02-15T19:53:11.284213Z"Grudka, Andrzej"https://zbmath.org/authors/?q=ai:grudka.andrzej"Kurzyński, Paweł"https://zbmath.org/authors/?q=ai:kurzynski.pawel"Polak, Tomasz P."https://zbmath.org/authors/?q=ai:polak.tomasz-p"Sajna, Adam S."https://zbmath.org/authors/?q=ai:sajna.adam-s"Wójcik, Jan"https://zbmath.org/authors/?q=ai:wojcik.jan"Wójcik, Antoni"https://zbmath.org/authors/?q=ai:wojcik.antoniSummary: The eigenbases of two quantum observables, \(\{|a_i\rangle\}^D_{i=1}\) and \(\{|b_j\rangle\}^D_{j=1}\), form mutually unbiased bases (MUB) if \(|\langle a_i|b_j\rangle| = 1/\sqrt{D}\) for all \(i\) and \(j\). In realistic situations MUB are hard to obtain and one looks for approximate MUB (AMUB), in which case the corresponding eigenbases obey \(|\langle a_i|b_j\rangle|\leqslant c/\sqrt{D}\), where \(c\) is some positive constant independent of \(D\). In majority of cases observables corresponding to MUB and AMUB do not have clear physical interpretation. Here we study discrete-time quantum walks (QWs) on \(d\)-cycles with a position and coin-dependent phase-shift. Such a model simulates a dynamics of a quantum particle moving on a ring with an artificial gauge field. In our case the amplitude of the phase-shift is governed by a single discrete parameter \(q\). We solve the model analytically and observe that for prime \(d\) the eigenvectors of two QW evolution operators form AMUB. Namely, if \(d\) is prime the corresponding eigenvectors of the evolution operators, that act in the \(D\)-dimensional Hilbert space (\(D = 2d\)), obey \(|\langle v_q|v^\prime_{q^\prime}\rangle| \leqslant \sqrt{2}/\sqrt{D}\) for \(q \neq q^\prime\) and for all \(|v_q\rangle\) and \(|v^\prime_{q^\prime}\rangle\). Finally, we show that the analogous AMUB relation still holds in the continuous version of this model, which corresponds to a one-dimensional Dirac particle.Dynamical behavior of quantum correlation entropy under the noisy quantum channel for multiqubit systemshttps://zbmath.org/1526.810042024-02-15T19:53:11.284213Z"Zhou, Xiang"https://zbmath.org/authors/?q=ai:zhou.xiang.1|zhou.xiangSummary: Quantum correlation entropy is used to measure total non-classical correlation of multiple states. It is based on a local coarse-grained measurement. Quantum noisy processes have a theoretically and experimentally important role in quantum information tasks. We study the dynamical behavior of quantum correlation entropy of output state under the effected of the \textit{bit}-\textit{flip} channel, \textit{phase}-\textit{flip} channel and \textit{bit}-\textit{phase} flip channel. We find that quantum correlation entropy of output state exists the frozen phenomenon and the phenomenon of sudden death. The phenomenon of sudden death shows that under the influence of a noisy channel, the output state becomes a classically correlated state.Nonclassical properties of a hybrid NAAN quantum statehttps://zbmath.org/1526.810072024-02-15T19:53:11.284213Z"Ren, Gang"https://zbmath.org/authors/?q=ai:ren.gang"Yu, Hai-jun"https://zbmath.org/authors/?q=ai:yu.haijun"Zhang, Chun-zao"https://zbmath.org/authors/?q=ai:zhang.chun-zao"Chen, Feng"https://zbmath.org/authors/?q=ai:chen.feng.1Summary: We introduce a two-mode hybrid entangled state (NAAN) which is constructed by two n-photon Fock states and two coherent states with an arbitrary relative phase. We show that the NAAN can be considered as the superpositions of NOON states when \(\alpha \neq 0\). In the special case, when \(\alpha = 0\), the NAAN degenerates to the general NOON. The most interesting nonclassical properties of this state are its strong violations of the CHSH inequality. In addition, we show explicitly some typical nonclassical properties of the NAAN state, such as entanglement, sub-Poissonian distribution, phase fluctuation and squeezing. These findings suggest that the even NAAN states exhibit a high degree of entanglement, while the odd NAAN states have a distinct sub-Poissonian distribution and the optimal phase sensitivity.Asymmetric controlled quantum teleportation via eight-qubit entangled state in a noisy environmenthttps://zbmath.org/1526.810132024-02-15T19:53:11.284213Z"Kaur, Simranjot"https://zbmath.org/authors/?q=ai:kaur.simranjot"Gill, Savita"https://zbmath.org/authors/?q=ai:gill.savitaSummary: Using eight qubit entangled state, we present a unique protocol for controlled quantum teleportation of one and two-qubit state. This protocol has five parties, one sender Alice who wants to teleport her two-qubit state to receiver Bob, and one qubit state to two distant receivers Charlie and David under the supervision of Eve. The entangled state of eight qubits acts as a channel to connect sender and receivers. In this protocol, we have reported a controlled teleportation protocol in which a single user can teleport different information to three receivers under the controller. This protocol is based on Bell state measurement (BSM), GHZ state measurement (GHZ), single-qubit measurement (SM), and unitary operations (UO). Further, the protocol is proposed in four different noisy channels including the phase-flip noise (PF), bit-flip noise (BF), amplitude-damping noise (ADN), and phase-damping noise (PDN) and it is found that it only depends on the amplitude coefficient of an initial state and noise intensity.A verifiable \((k,n)\)-threshold quantum secure multiparty summation protocolhttps://zbmath.org/1526.810212024-02-15T19:53:11.284213Z"Li, Fulin"https://zbmath.org/authors/?q=ai:li.fulin"Hu, Hang"https://zbmath.org/authors/?q=ai:hu.hang"Zhu, Shixin"https://zbmath.org/authors/?q=ai:zhu.shixin"Li, Ping"https://zbmath.org/authors/?q=ai:li.ping.2Summary: Quantum secure multiparty summation plays an important role in quantum cryptography. In the existing quantum secure multiparty summation protocols, the \((n,n)\)-threshold protocol has been given extensive attention. To increase the applicability of quantum secure multiparty summation protocols, a new quantum secure multiparty summation protocol based on Shamir's threshold scheme and \(d\)-dimensional GHZ state is proposed in this paper. In the proposed protocol, \textit{i)} it has a \((k,n)\)-threshold approach; \textit{ii)} in the result output phase, it can not only detect the existence of deceptive behavior but also determine the specific cheaters; \textit{iii)} compared with the \((n,n)\)-threshold quantum secure multiparty summation protocols, it needs less computation cost when \(L\) satisfies \(L > 4\), where \(L\) is the length of each participant's secret. In addition, the security analysis shows that our protocol can resist intercept-resend attack, entangle-measure attack, Trojan horse attack, and participant attack.Visually meaningful quantum color image encryption scheme based on measured alternate quantum walks and quantum logistic mixed linear-nonlinear coupled mapping latticeshttps://zbmath.org/1526.810232024-02-15T19:53:11.284213Z"Yu, Fang-Fang"https://zbmath.org/authors/?q=ai:yu.fang-fang"Dai, Jing-Yi"https://zbmath.org/authors/?q=ai:dai.jingyi"Liu, Si-Hang"https://zbmath.org/authors/?q=ai:liu.si-hang"Gong, Li-Hua"https://zbmath.org/authors/?q=ai:gong.lihuaSummary: The probability distribution generated by quantum walk may appear zero-probability at multiple positions. To avoid this limitation, the measured alternate quantum walks (MAQWs) are presented. A quantum color image encryption algorithm is presented based on the MAQWs and the quantum logistic mixed linear-nonlinear coupled mapping lattices (QLMLNCML). The plaintext image is transformed into a quantum form with quantum representation for color digital images (NCQI). Subsequently, the quantum image is divided into two sub-blocks, on which permutation and diffusion operations are respectively implemented with the probability distribution generated by the MAQWs. To achieve high randomness and high complexity, the QLMLNCML is obtained by combining the quantum logistic map with the mixed linear-nonlinear coupled mapping lattices. The two encryption sub-blocks are combined and then XORed with the quantum key images originated from the QLMLNCML to acquire the secret image. The whole encryption scheme employs the basic quantum gates, including C-NOT gate, swap gate and Toffoli gate. To further enhance the security of the cryptosystem, a novel visually meaningful image encryption algorithm is designed, and the generated visually meaningful ciphertext image is more imperceptible than the secret image. The proposed quantum color image encryption algorithm is superior to its classical counterparts in terms of security, robustness and computational complexity.Heisenberg uncertainty principle: an advanced undergraduate laboratory experiment based on quantum quadrature operatorshttps://zbmath.org/1526.810342024-02-15T19:53:11.284213Z"Chen, Yu"https://zbmath.org/authors/?q=ai:chen.yu.16|chen.yuqun"Liu, Zhaoyang"https://zbmath.org/authors/?q=ai:liu.zhaoyang"Chen, Xin"https://zbmath.org/authors/?q=ai:chen.xin.9|chen.xin.2|chen.xin.3|chen.xin.1"Liu, Shuqi"https://zbmath.org/authors/?q=ai:liu.shuqi"Jiang, Jiatong"https://zbmath.org/authors/?q=ai:jiang.jiatong"Wu, Yuan"https://zbmath.org/authors/?q=ai:wu.yuan"Guo, Jinxian"https://zbmath.org/authors/?q=ai:guo.jinxian"Chen, L. Q."https://zbmath.org/authors/?q=ai:chen.liuqing|chen.liqing|chen.liqian|chen.liqun.1|chen.linqiang|chen.liqiong|chen.longqing|chen.lanqing|chen.liangquan|chen.liquan|chen.liqiang|chen.lingqiao|chen.liangqun|chen.liqun|chen.lin-qing|chen.lingqiang(no abstract)From statistical physics to data-driven modelling. With applications to quantitative biologyhttps://zbmath.org/1526.820012024-02-15T19:53:11.284213Z"Cocco, Simona"https://zbmath.org/authors/?q=ai:cocco.simona"Monasson, Rémi"https://zbmath.org/authors/?q=ai:monasson.remi"Zamponi, Francesco"https://zbmath.org/authors/?q=ai:zamponi.francescoPublisher's description: The study of most scientific fields now relies on an ever-increasing amount of data, due to instrumental and experimental progress in monitoring and manipulating complex systems made of many microscopic constituents. How can we make sense of such data, and use them to enhance our understanding of biological, physical, and chemical systems?
Aimed at graduate students in physics, applied mathematics, and computational biology, the primary objective of this textbook is to introduce the concepts and methods necessary to answer this question at the intersection of probability theory, statistics, optimisation, statistical physics, inference, and machine learning.
The second objective of this book is to provide practical applications for these methods, which will allow students to assimilate the underlying ideas and techniques. While readers of this textbook will need basic knowledge in programming (Python or an equivalent language), the main emphasis is not on mathematical rigour, but on the development of intuition and the deep connections with statistical physics.Optical solitons in curved spacetimehttps://zbmath.org/1526.830142024-02-15T19:53:11.284213Z"Spengler, Felix"https://zbmath.org/authors/?q=ai:spengler.felix"Belenchia, Alessio"https://zbmath.org/authors/?q=ai:belenchia.alessio"Rätzel, Dennis"https://zbmath.org/authors/?q=ai:ratzel.dennis"Braun, Daniel"https://zbmath.org/authors/?q=ai:braun.daniel|braun.daniel-aSummary: Light propagation in curved spacetime is at the basis of some of the most stringent tests of Einstein's general relativity. At the same time, light propagation in media is at the basis of several communication systems. Given the ubiquity of the gravitational field, and the exquisite level of sensitivity of optical measurements, the time is ripe for investigations combining these two aspects and studying light propagation in media located in curved spacetime. In this work, we focus on the effect of a weak gravitational field on the propagation of optical solitons in non-linear optical media. We derive a non-linear Schrödinger equation describing the propagation of an optical pulse in an effective, gradient-index medium in flat spacetime, encoding both the material properties and curved spacetime effects. In analyzing the special case of propagation in a 1D optical fiber, we also include the effect of mechanical deformations and show it to be the dominant effect for a fiber oriented in the radial direction in Schwarzschild spacetime.The optimal dynamic rationing policy in the stock-rationing queuehttps://zbmath.org/1526.900122024-02-15T19:53:11.284213Z"Li, Quan-Lin"https://zbmath.org/authors/?q=ai:li.quanlin"Li, Yi-Meng"https://zbmath.org/authors/?q=ai:li.yi-meng"Ma, Jing-Yu"https://zbmath.org/authors/?q=ai:ma.jingyu"Liu, Heng-Li"https://zbmath.org/authors/?q=ai:liu.heng-liSummary: In this paper, we study a stock-rationing queue with two demand classes by means of the sensitivity-based optimization, and develop a complete algebraic solution for the optimal dynamic rationing policy. To do this, we establish a policy-based birth-death process to show that the optimal dynamic rationing policy must be of transformational threshold type. Based on this finding, we can refine three sufficient conditions under each of which the optimal dynamic rationing policy is of threshold type (i.e., critical rationing level). Crucially, we characterize the monotonicity and optimality of the long-run average profit of this system, and establish some new structural properties of the optimal dynamic rationing policy by observing any given reference policy. Finally, we use numerical examples to verify computability of our theoretical results. We believe that the methodology and results developed in this paper can shed light on the study of stock-rationing queue and open a series of potentially promising research.
For the entire collection see [Zbl 1514.68006].Stability of a cascade system with two stations and its extension for multiple stationshttps://zbmath.org/1526.900132024-02-15T19:53:11.284213Z"Miyazawa, Masakiyo"https://zbmath.org/authors/?q=ai:miyazawa.masakiyo"Morozov, Evsey"https://zbmath.org/authors/?q=ai:morozov.evseySummary: We consider a two-station cascade system in which waiting or externally arriving customers at station 1 move to the station 2 if the queue size of station 1 including an arriving customer itself and a customer being served is greater than a given threshold level \(c_1\ge 1\) and if station 2 is empty. Assuming that external arrivals are subject to independent renewal processes satisfying certain regularity conditions and service times are i.i.d. at each station, we derive necessary and sufficient conditions for a Markov process describing this system to be positive recurrent in the sense of Harris. This result is extended to the cascade system with a general number \(k\) of stations in series. This extension requires certain traffic intensities of stations \(2,3,\dots,k-1\) for \(k\ge 3\) to be defined. We finally note that the modeling assumptions on the renewal arrivals and \(i.i.d\). service times are not essential if the notion of the stability is replaced by a certain sample path condition. This stability notion is identical with the standard stability if the whole system is described by the Markov process which is a Harris irreducible \(T\)-process.Household investment-consumption-insurance policies under the age-dependent risk preferenceshttps://zbmath.org/1526.910052024-02-15T19:53:11.284213Z"Wang, Hao"https://zbmath.org/authors/?q=ai:wang.hao.31"Wang, Ning"https://zbmath.org/authors/?q=ai:wang.ning.2"Xu, Lin"https://zbmath.org/authors/?q=ai:xu.lin.3"Hu, Shujie"https://zbmath.org/authors/?q=ai:hu.shujie"Yan, Xingyu"https://zbmath.org/authors/?q=ai:yan.xingyuSummary: In this paper, we examine the optimal investment-consumption-insurance policies for a wage earner with time-varying risk preferences. The wage earner's objective is to find the optimal investment-consumption-insurance strategies that maximise the expected discounted utilities from intertemporal consumption, legacy and terminal wealth over the uncertain lifetime horizon. Similar to \textit{A. Lichtenstern} et al. [Math. Financ. Econ. 15, No. 2, 275--313 (2021; Zbl 1460.91229)], by using a separation approach, the problem is divided into two sub-problems, including the consumption-legacy problem and the terminal wealth-only problem. For each sub-problem, the analytical expressions for the optimal strategies and value functions are derived by using the martingale method. In such a way, we obtain the optimal strategies for the original problem by merging the solutions of the two individual problems. Finally, we conduct some numerical experiments to illustrate the effects of some parameters on the optimal strategies and obtain some economic insights.On one approach to mathematical modeling of socio-economic development of regionshttps://zbmath.org/1526.910172024-02-15T19:53:11.284213Z"Zherelo, A."https://zbmath.org/authors/?q=ai:zherelo.anatoly-v"Chernogorsky, S."https://zbmath.org/authors/?q=ai:chernogorsky.s"Shvetsov, K."https://zbmath.org/authors/?q=ai:shvetsov.konstantin(no abstract)Mean-variance portfolio selection under no-shorting rules: a BSDE approachhttps://zbmath.org/1526.910242024-02-15T19:53:11.284213Z"Zhang, Liangquan"https://zbmath.org/authors/?q=ai:zhang.liangquan"Li, Xun"https://zbmath.org/authors/?q=ai:li.xunSummary: This paper revisits the mean-variance portfolio selection problem in continuous-time within the framework of short-selling of stocks is prohibited via backward stochastic differential equation approach. To \textit{relax} the strong condition in [\textit{X. Li} et al., SIAM J. Control Optim. 40, No. 5, 1540--1555 (2002; Zbl 1027.91040)], the above issue is formulated as a stochastic recursive optimal linear-quadratic control problem. Due to no-shorting rules (namely, the portfolio taking non-negative values), the well-known ``completion of squares'' no longer applies directly. To overcome this difficulty, we study the corresponding Hamilton-Jacobi-Bellman (HJB, for short) equation inherently and derive the two groups of Riccati equations. On one hand, the value function constructed via Riccati equations is shown to be a viscosity solution of the HJB equation mentioned before; On the other hand, by means of these Riccati equations and backward semigroup, we are able to get explicitly the efficient frontier and efficient investment strategies for the recursive utility mean-variance portfolio optimization problem.Utility maximization in a stochastic affine interest rate and CIR risk premium framework: a BSDE approachhttps://zbmath.org/1526.910252024-02-15T19:53:11.284213Z"Zhang, Yumo"https://zbmath.org/authors/?q=ai:zhang.yumoThis paper studies the problem of portfolio optimization in a market with stochastic interest rates, for agents reporting expected utility preferences, within the framework of affine models for the underlying processes using the methodology of backward stochastic differential equations (BSDEs). This approach allows for the study of the non Markovian case, as well as cases such as the Heston or the Vacisek model. The analysis focuses on the case of the power and the logarithmic utility functions. For both cases the corresponding BSDE that characterizes the optimal strategy is constructed and particular solutions are obtained in terms of auxiliary ODE systems that are explicitly solved. Under certain assumptions it is shown that the obtained solutions are the unique solutions of the corresponding BSDE. The BSDE related to the power law utility case is nonlinear (and explicitly solvable upon reduction to a set of nonlinear ODEs reducible to the Riccatti equation) while for logarithmic utility case it is linear (and explicitly solvable upon reduction to a set of linear ODEs). The paper concludes with a numerical illustration of the results.
Reviewer: Athanasios Yannacopoulos (Athína)An implicit scheme for American put optionshttps://zbmath.org/1526.910302024-02-15T19:53:11.284213Z"Chen, Xinfu"https://zbmath.org/authors/?q=ai:chen.xinfu"Lu, Zhengyang"https://zbmath.org/authors/?q=ai:lu.zhengyang"Ma, Jingtang"https://zbmath.org/authors/?q=ai:ma.jingtang"Shen, Jinye"https://zbmath.org/authors/?q=ai:shen.jinyeSummary: In this paper, an implicit scheme is proposed to solve a parabolic variational inequality arising from the American put options. The discretization leads to a class of discrete elliptic variational inequalities. Well-posedness, including existence, uniqueness, comparison principle, and stability of the discrete elliptic variational inequality is established. A simple and efficient algorithm to solve the implicit discretized variational inequality is discovered. The novelty here is an explicit formula for the optimal exercise boundary. An improved algorithm is also presented to eliminate the singularity near the time to expiry. Numerical examples are carried out to show the accuracy and efficiency of the proposed algorithms.Fixation probability in evolutionary dynamics on switching temporal networkshttps://zbmath.org/1526.920332024-02-15T19:53:11.284213Z"Bhaumik, Jnanajyoti"https://zbmath.org/authors/?q=ai:bhaumik.jnanajyoti"Masuda, Naoki"https://zbmath.org/authors/?q=ai:masuda.naokiSummary: Population structure has been known to substantially affect evolutionary dynamics. Networks that promote the spreading of fitter mutants are called amplifiers of selection, and those that suppress the spreading of fitter mutants are called suppressors of selection. Research in the past two decades has found various families of amplifiers while suppressors still remain somewhat elusive. It has also been discovered that most networks are amplifiers of selection under the birth-death updating combined with uniform initialization, which is a standard condition assumed widely in the literature. In the present study, we extend the birth-death processes to temporal (i.e., time-varying) networks. For the sake of tractability, we restrict ourselves to switching temporal networks, in which the network structure deterministically alternates between two static networks at constant time intervals or stochastically in a Markovian manner. We show that, in a majority of cases, switching networks are less amplifying than both of the two static networks constituting the switching networks. Furthermore, most small switching networks, i.e., networks on six nodes or less, are suppressors, which contrasts to the case of static networks.Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equationshttps://zbmath.org/1526.920352024-02-15T19:53:11.284213Z"Champagnat, Nicolas"https://zbmath.org/authors/?q=ai:champagnat.nicolas"Méléard, Sylvie"https://zbmath.org/authors/?q=ai:meleard.sylvie"Mirrahimi, Sepideh"https://zbmath.org/authors/?q=ai:mirrahimi.sepideh"Tran, Viet Chi"https://zbmath.org/authors/?q=ai:tran.viet-chiSummary: We consider a stochastic model for the evolution of a discrete population structured by a trait with values on a finite grid of the torus, and with mutation and selection. We focus on a parameter scaling where population is large, individual mutations are small but not rare, and the grid mesh is much smaller than the size of mutation steps. When considering the evolution of the population in long time scales, the contribution of small sub-populations may strongly influence the dynamics. Our main result quantifies the asymptotic dynamics of subpopulation sizes on a logarithmic scale. We establish that under our rescaling, the stochastic discrete process converges to the viscosity solution of a Hamilton-Jacobi equation. The proof makes use of almost sure maximum principles and careful control of the martingale parts.Numerical stochastic modeling of dynamics of interacting populationshttps://zbmath.org/1526.920492024-02-15T19:53:11.284213Z"Pertsev, N. V."https://zbmath.org/authors/?q=ai:pertsev.nikolai-viktorovich"Topchiĭ, V. A."https://zbmath.org/authors/?q=ai:topchii.valentin"Loginov, K. K."https://zbmath.org/authors/?q=ai:loginov.konstantin-konstantinovich(no abstract)Temporal and probabilistic comparisons of epidemic interventionshttps://zbmath.org/1526.920522024-02-15T19:53:11.284213Z"Boudreau, Mariah C."https://zbmath.org/authors/?q=ai:boudreau.mariah-c"Allen, Andrea J."https://zbmath.org/authors/?q=ai:allen.andrea-j"Roberts, Nicholas J."https://zbmath.org/authors/?q=ai:roberts.nicholas-j"Allard, Antoine"https://zbmath.org/authors/?q=ai:allard.antoine"Hébert-Dufresne, Laurent"https://zbmath.org/authors/?q=ai:hebert-dufresne.laurentSummary: Forecasting disease spread is a critical tool to help public health officials design and plan public health interventions. However, the expected future state of an epidemic is not necessarily well defined as disease spread is inherently stochastic, contact patterns within a population are heterogeneous, and behaviors change. In this work, we use time-dependent probability generating functions (PGFs) to capture these characteristics by modeling a stochastic branching process of the spread of a disease over a network of contacts in which public health interventions are introduced over time. To achieve this, we define a general transmissibility equation to account for varying transmission rates (e.g. masking), recovery rates (e.g. treatment), contact patterns (e.g. social distancing) and percentage of the population immunized (e.g. vaccination). The resulting framework allows for a temporal and probabilistic analysis of an intervention's impact on disease spread, which match continuous-time stochastic simulations that are much more computationally expensive. To aid policy making, we then define several metrics over which temporal and probabilistic intervention forecasts can be compared: Looking at the expected number of cases and the worst-case scenario over time, as well as the probability of reaching a critical level of cases and of not seeing any improvement following an intervention. Given that epidemics do not always follow their average expected trajectories and that the underlying dynamics can change over time, our work paves the way for more detailed short-term forecasts of disease spread and more informed comparison of intervention strategies.Mathematical modeling of smoking habits in the societyhttps://zbmath.org/1526.920562024-02-15T19:53:11.284213Z"Sofia, I. R."https://zbmath.org/authors/?q=ai:sofia.i-r"Ghosh, Mini"https://zbmath.org/authors/?q=ai:ghosh.miniSummary: In this study, we formulate and analyze a non-linear mathematical model to study the dynamics of smoking and its impact on society. This is a compartment model which has four compartments, namely, potential smoker, occasional smoker, smoker and quitters. As per WHO, each year there is a significant number of smoking-related deaths. Keeping this in view, we have incorporated smoking-related death in our proposed model. Further, we investigate the model for possible equilibria, compute the basic reproduction number and investigate the stability of obtained equilibria. Later we extend this model to stochastic model and perform numerical simulation for both the deterministic and the stochastic model. The results of stochastic model are almost similar to the results obtained for deterministic model. We have also explored the impact of the parameters related to quitting smoking habits on the equilibrium level of occasional smokers. We also perform sensitivity analysis to find the key parameters which make significant change in the reproduction numbers.On the practical stability with regard to a part of the variables for distribution-dependent SDEs driven by time-changed Brownian motionhttps://zbmath.org/1526.931962024-02-15T19:53:11.284213Z"Li, Xiaocong"https://zbmath.org/authors/?q=ai:li.xiaocong"Ren, Yong"https://zbmath.org/authors/?q=ai:ren.yong.1|ren.yong.2|ren.yongSummary: Distribution-dependent stochastic differential equations (DDSDEs, in short), also called as McKean-Vlasov SDEs, are a special class of SDEs and have some concrete applications in mean-field control, mean-field games and complex networks. Stability plays fundamental role among the theory and applications for DDSDEs. In this paper, we discuss a class of distribution-dependent stochastic differential equations driven by time-changed Brownian motions (TDDSDEs, in short). The main aim of this paper is to establish the stability for TDDSDEs. Sufficient conditions are provided to guarantee the solutions to be stable in different senses with regard to a part of the variables in terms of a Lyapunov function. Two examples are given to show the usefulness of the obtained theoretical results.Synchronization for stochastic coupled networks with Lévy noise via event-triggered controlhttps://zbmath.org/1526.932652024-02-15T19:53:11.284213Z"Dong, Hailing"https://zbmath.org/authors/?q=ai:dong.hailing"Luo, Ming"https://zbmath.org/authors/?q=ai:luo.ming.1"Xiao, Mingqing"https://zbmath.org/authors/?q=ai:xiao.mingqingSummary: This paper addresses the realization of almost sure synchronization problem for a new array of stochastic networks associated with delay and Lévy noise via event-triggered control. The coupling structure of the network is governed by a continuous-time homogeneous Markov chain. The nodes in the networks communicate with each other and update their information only at discrete-time instants so that the network workload can be minimized. Under the framework of stochastic process including Markov chain and Lévy process, and the convergence theorem of non-negative semi-martingales, we show that the Markovian coupled networks can achieve the almost sure synchronization by event-triggered control methodology. The results are further extended to the directed topology, where the coupling structure can be asymmetric. Furthermore, we also proved that the Zeno behavior can be excluded under our proposed approach, indicating that our framework is practically feasible. Numerical simulations are provided to demonstrate the effectiveness of the obtained theoretical results.A novel algorithm for asymptotic stability analysis of some classes of stochastic time-fractional Volterra equationshttps://zbmath.org/1526.932692024-02-15T19:53:11.284213Z"Ponosov, Arcady"https://zbmath.org/authors/?q=ai:ponosov.arkadii-vladimirovich"Idels, Lev"https://zbmath.org/authors/?q=ai:idels.lev-v"Kadiev, Ramazan I."https://zbmath.org/authors/?q=ai:kadiev.ramazan-ismailovichSummary: The article presents a regularization method to study global asymptotic stability of stochastic multi-time scale Volterra equations as an alternative to the algorithms based on Lyapunov functionals. Two different concepts of stability were linked and examined. The central idea of the method is based on a parallelism between the Lyapunov stability and a stochastic version of the input-to-state stability, which is well-known in the control theory of deterministic ordinary differential equations. At the first step the algorithm transforms the given delay equation into a Volterra differential equation with stochastic control, following replacement of the Lyapunov stability of the former with the input-to-state stability of the latter. To estimate the norms of the solutions we use a regularization technique based on the concept of inverse-positive matrices. This algorithm could be extended and applied for stability analysis of new classes of stochastic fractional differential equations and their applications.Pointwise second-order necessary conditions for stochastic optimal control with jump diffusionshttps://zbmath.org/1526.932782024-02-15T19:53:11.284213Z"Ghoul, Abdelhak"https://zbmath.org/authors/?q=ai:ghoul.abdelhak"Hafayed, Mokhtar"https://zbmath.org/authors/?q=ai:hafayed.mokhtar"Lakhdari, Imad Eddine"https://zbmath.org/authors/?q=ai:lakhdari.imad-eddine"Meherrem, Shahlar"https://zbmath.org/authors/?q=ai:meherrem.shahlarSummary: In this paper, we establish a second-order necessary conditions for stochastic optimal control for jump diffusions. The controlled system is described by a stochastic differential systems driven by Poisson random measure and an independent Brownian motion. The control domain is assumed to be convex. Pointwise second-order maximum principle for controlled jump diffusion in terms of the martingale with respect to the time variable is proved. The proof of the main result is based on variational approach using the stochastic calculus of jump diffusions and some estimates on the state processes.Risk-sensitive maximum principle for stochastic optimal control of mean-field type Markov regime-switching jump-diffusion systemshttps://zbmath.org/1526.932802024-02-15T19:53:11.284213Z"Moon, Jun"https://zbmath.org/authors/?q=ai:moon.jun-heeSummary: We consider the risk-sensitive optimal control problem for mean-field type Markov regime-switching jump-diffusion systems driven by Brownian motions and Poisson jumps with (Markovian) switching coefficients. The system is coupled with its mean-filed term, that is, the expected value of the state process, and the objective functional is of the risk-sensitive type. Our problem is closely related to the mean-field type robust optimization problem for a general class of stochastic jump systems due to the inherent feature of the risk-sensitive objective functional. By establishing the logarithmic transformations of the associated equivalent singular risk-neutral control problem, we obtain the risk-sensitive maximum principle type necessary and sufficient conditions for optimality, where the sufficient condition requires an additional convexity assumption. The risk-sensitive maximum principle in this article is characterized as the variational inequality, together with the first- and second-order (mean-field type) adjoint processes as well as the auxiliary first-order adjoint process. Unlike the risk-neutral and mean-field free cases, the additional adjoint equation is induced due to the mean-field coupling term and the risk-sensitive logarithmic transformation. We apply the risk-sensitive maximum principle of this article to the risk-sensitive linear-quadratic problem, for which an explicit optimal solution is obtained.
{{\copyright} 2021 John Wiley \& Sons Ltd.}Entropy-variance inequalities for discrete log-concave random variables via degree of freedomhttps://zbmath.org/1526.940112024-02-15T19:53:11.284213Z"Aravinda, Heshan"https://zbmath.org/authors/?q=ai:aravinda.heshanLet \(X\) be an integer-valued random variable with contiguous support, and suppose that \(X\) has a log-concave mass function, that is, \(\mathbb{P}(X=x)^2\geq\mathbb{P}(X=x-1)\mathbb{P}(X=x+1)\) for all \(x\). For \(\alpha>0\) and \(\alpha\not=1\), let
\[
H_{\alpha}(X)=\frac{1}{1-\alpha}\log\sum_{x}\mathbb{P}(X=x)^\alpha
\]
be the Rényi entropy of order \(\alpha\), with the limit as \(\alpha\to\infty\) the min-entropy \(H_\infty(X)=-\log\max_x\mathbb{P}(X=x)\), and the limit as \(\alpha\to1\) being the usual Shannon entropy. The author shows that the entropy power \(N_\alpha(X)=e^{2H_\alpha(X)}\) satisfies
\[
N_\infty(X)\geq1+\mbox{Var}(X)\,,
\]
with equality asymptotically achieved for a geometric distribution with parameter going to either 0 or 1. This result is used to establish the following inequality for entropy power: if \(S_n=X_1+\cdots+X_n\) is a sum of independent discrete log-concave random variables as above, then
\[
\Delta_\alpha(S_n)\geq\frac{\alpha-1}{4(3\alpha-1)}\sum_{i=1}^n\Delta_\alpha(X_i)
\]
for \(\alpha>1\), where \(\Delta_{\alpha}(X)=N_\alpha(X)-1\). The proof of the main result makes use of a definition of the degree of freedom of a log-concave sequence.
Reviewer: Fraser Daly (Edinburgh)Random feedback shift registers and the limit distribution for largest cycle lengthshttps://zbmath.org/1526.940162024-02-15T19:53:11.284213Z"Arratia, Richard A."https://zbmath.org/authors/?q=ai:arratia.richard"Canfield, E. Rodney"https://zbmath.org/authors/?q=ai:canfield.e-rodney"Hales, Alfred W."https://zbmath.org/authors/?q=ai:hales.alfred-wSummary: For a random binary noncoalescing feedback shift register of width \(n\), with all \(2^{2^{n-1}}\) possible feedback functions \(f\) equally likely, the process of long cycle lengths, scaled by dividing by \(N=2^n\), converges in distribution to the same Poisson-Dirichlet limit as holds for random permutations in \(\mathcal{S}_N\), with all \(N!\) possible permutations equally likely. Such behaviour was conjectured by \textit{S. W. Golomb} et al. [Cycles from nonlinear feedback shift registers. Techn. Rep., Pasadena, CA: Jet Propulsion Laboratory, California Institute for Technology (1959)].