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The Annals of Applied Probability
The Annals of Applied Probability
Deaconu, Madalina; Herrmann, Samuel
The initial-boundary value problem for the heat equation is solved by using an algorithm based on a random walk on heat balls. Even if it represents a sophisticated generalization of the Walk on Spheres (WOS) algorithm introduced to solve the Dirichlet problem for Laplace’s equation, its implementation is rather easy. The construction of this algorithm can be considered as a natural consequence of previous works the authors completed on the hitting time approximation for Bessel processes and Brownian motion [Ann. Appl. Probab. 23 (2013) 2259–2289, Math. Comput. Simulation 135 (2017) 28–38, Bernoulli 23 (2017) 3744–3771]. A similar procedure was introduced...
Hening, Alexandru; Nguyen, Dang H.
In recent years there has been a growing interest in the study of the dynamics of stochastic populations. A key question in population biology is to understand the conditions under which populations coexist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of $n$ populations that live in a stochastic environment and which can interact nonlinearly (through competition for resources, predator–prey behavior, etc.). Our models are described by $n$-dimensional Kolmogorov systems with white noise (stochastic differential equations—SDE). We give sharp conditions under...
Blanchard, Romain; Carassus, Laurence
This paper investigates the problem of maximizing expected terminal utility in a discrete-time financial market model with a finite horizon under nondominated model uncertainty. We use a dynamic programming framework together with measurable selection arguments to prove that under mild integrability conditions, an optimal portfolio exists for an unbounded utility function defined on the half-real line.
Agazzi, Andrea; Dembo, Amir; Eckmann, Jean-Pierre
We prove a sample path Large Deviation Principle (LDP) for a class of jump processes whose rates are not uniformly Lipschitz continuous in phase space. Building on it, we further establish the corresponding Wentzell–Freidlin (W-F) (infinite time horizon) asymptotic theory. These results apply to jump Markov processes that model the dynamics of chemical reaction networks under mass action kinetics, on a microscopic scale. We provide natural sufficient topological conditions for the applicability of our LDP and W-F results. This then justifies the computation of nonequilibrium potential and exponential transition time estimates between different attractors in the large volume limit, for...
Pitman, Jim; Yakubovich, Yuri
We describe the distribution of frequencies ordered by sample values in a random sample of size $n$ from the two parameter $\mathsf{GEM}(\alpha,\theta)$ random discrete distribution on the positive integers. These frequencies are a (size-$\alpha$)-biased random permutation of the sample frequencies in either ranked order, or in the order of appearance of values in the sampling process. This generalizes a well-known identity in distribution due to Donnelly and Tavaré [Adv. in Appl. Probab. 18 (1986) 1–19] for $\alpha=0$ to the case $0\le\alpha<1$. This description extends to sampling from $\operatorname{Gibbs}(\alpha)$ frequencies obtained by suitable conditioning of the $\mathsf{GEM}(\alpha,\theta)$ model, and yields a...
van der Hoorn, Pim; Olvera-Cravioto, Mariana
We analyze the distribution of the distance between two nodes, sampled uniformly at random, in digraphs generated via the directed configuration model, in the supercritical regime. Under the assumption that the covariance between the in-degree and out-degree is finite, we show that the distance grows logarithmically in the size of the graph. In contrast with the undirected case, this can happen even when the variance of the degrees is infinite. The main tool in the analysis is a new coupling between a breadth-first graph exploration process and a suitable branching process based on the Kantorovich–Rubinstein metric. This coupling holds uniformly...
Ding, Xiucai; Yang, Fan
In this paper, we prove a necessary and sufficient condition for the edge universality of sample covariance matrices with general population. We consider sample covariance matrices of the form $\mathcal{Q}=TX(TX)^{*}$, where $X$ is an $M_{2}\times N$ random matrix with $X_{ij}=N^{-1/2}q_{ij}$ such that $q_{ij}$ are i.i.d. random variables with zero mean and unit variance, and $T$ is an $M_{1}\times M_{2}$ deterministic matrix such that $T^{*}T$ is diagonal. We study the asymptotic behavior of the largest eigenvalues of $\mathcal{Q}$ when $M:=\min\{M_{1},M_{2}\}$ and $N$ tend to infinity with $\lim_{N\to\infty}{N}/{M}=d\in(0,\infty)$. We prove that the Tracy–Widom law holds for the largest eigenvalue of $\mathcal{Q}$ if...
Bandini, Elena; Cosso, Andrea; Fuhrman, Marco; Pham, Huyên
We introduce a suitable backward stochastic differential equation (BSDE) to represent the value of an optimal control problem with partial observation for a controlled stochastic equation driven by Brownian motion. Our model is general enough to include cases with latent factors in mathematical finance. By a standard reformulation based on the reference probability method, it also includes the classical model where the observation process is affected by a Brownian motion (even in presence of a correlated noise), a case where a BSDE representation of the value was not available so far. This approach based on BSDEs allows for greater generality...
Thacker, Debleena; Volkov, Stanislav
Start with a graph with a subset of vertices called the border. A particle released from the origin performs a random walk on the graph until it comes to the immediate neighbourhood of the border, at which point it joins this subset thus increasing the border by one point. Then a new particle is released from the origin and the process repeats until the origin becomes a part of the border itself. We are interested in the total number $\xi$ of particles to be released by this final moment.
¶ We show that this model covers the OK Corral model as...
Chhita, S.; Ferrari, P. L.; Spohn, H.
For stationary KPZ growth in $1+1$ dimensions, the height fluctuations are governed by the Baik–Rains distribution. Using the totally asymmetric single step growth model, alias TASEP, we investigate height fluctuations for a general class of spatially homogeneous random initial conditions. We prove that for TASEP there is a one-parameter family of limit distributions, labeled by the diffusion coefficient of the initial conditions. The distributions are defined through a variational formula. We use Monte Carlo simulations to obtain their numerical plots. Also discussed is the connection to the six-vertex model at its conical point.
Jagannath, Aukosh; Ko, Justin; Sen, Subhabrata
We study the asymptotic behavior of the Max $\kappa$-cut on a family of sparse, inhomogeneous random graphs. In the large degree limit, the leading term is a variational problem, involving the ground state of a constrained inhomogeneous Potts spin glass. We derive a Parisi-type formula for the free energy of this model, with possible constraints on the proportions, and derive the limiting ground state energy by a suitable zero temperature limit.
Swart, Jan M.
In 1964, G. J. Stigler introduced a stochastic model for the evolution of an order book on a stock market. This model was independently rediscovered and generalized by H. Luckock in 2003. In his formulation, traders place buy and sell limit orders of unit size according to independent Poisson processes with possibly different intensities. Newly arriving buy (sell) orders are either immediately matched to the best available matching sell (buy) order or stay in the order book until a matching order arrives. Assuming stationarity, Luckock showed that the distribution functions of the best buy and sell order in the order...
Hoze, Nathanael; Holcman, David
Coagulation-fragmentation processes describe the stochastic association and dissociation of particles in clusters. Cluster dynamics with cluster-cluster interactions for a finite number of particles has recently attracted attention especially in stochastic analysis and statistical physics of cellular biology, as novel experimental data are now available, but their interpretation remains challenging. We derive here probability distribution functions for clusters that can either aggregate upon binding to form clusters of arbitrary sizes or a single cluster can dissociate into two sub-clusters. Using combinatorics properties and Markov chain representation, we compute steady-state distributions and moments for the number of particles per cluster in the...
Battiston, Marco; Favaro, Stefano; Roy, Daniel M.; Teh, Yee Whye
We characterize the class of exchangeable feature allocations assigning probability $V_{n,k}\prod_{l=1}^{k}W_{m_{l}}U_{n-m_{l}}$ to a feature allocation of $n$ individuals, displaying $k$ features with counts $(m_{1},\ldots,m_{k})$ for these features. Each element of this class is parametrized by a countable matrix $V$ and two sequences $U$ and $W$ of nonnegative weights. Moreover, a consistency condition is imposed to guarantee that the distribution for feature allocations of $(n-1)$ individuals is recovered from that of $n$ individuals, when the last individual is integrated out. We prove that the only members of this class satisfying the consistency condition are mixtures of three-parameter Indian buffet Processes over...
Bella, Peter; Fehrman, Benjamin; Otto, Felix
We study the behavior of second-order degenerate elliptic systems in divergence form with random coefficients which are stationary and ergodic. Assuming moment bounds like Chiarini and Deuschel (2014) on the coefficient field $a$ and its inverse, we prove an intrinsic large-scale $C^{1,\alpha}$-regularity estimate for $a$-harmonic functions and obtain a first-order Liouville theorem for $a$-harmonic functions.
Chen, Wei-Kuo; Panchenko, Dmitry
We prove disorder chaos at zero temperature for three types of diluted models with large connectivity parameter: $K$-spin antiferromagnetic Ising model for even $K\geq2$, $K$-spin spin glass model for even $K\geq2$, and random $K$-sat model for all $K\geq2$. We show that modifying even a small proportion of clauses results in near maximizers of the original and modified Hamiltonians being nearly orthogonal to each other with high probability. We use a standard technique of approximating diluted models by appropriate fully connected models and then apply disorder chaos results in this setting, which include both previously known results as well as new...
Bernardin, Cédric; Gonçalves, Patrícia; Jara, Milton
We consider a chain of weakly harmonic coupled oscillators perturbed by a conservative noise. We show that by tuning accordingly the coupling constant, energy can diffuse like a Brownian motion or superdiffuse like a maximally $3/2$-stable asymmetric Lévy process. For a critical value of the coupling, the energy diffusion is described by a family of Lévy processes which interpolate between these two processes.
Guo, Heng; Jerrum, Mark
We show that the mixing time of Glauber (single edge update) dynamics for the random cluster model at $q=2$ on an arbitrary $n$-vertex graph is bounded by a polynomial in $n$. As a consequence, the Swendsen–Wang algorithm for the ferromagnetic Ising model at any temperature also has a polynomial mixing time bound.
Turova, Tatyana S.
We consider the system of particles on a finite interval with pairwise nearest neighbours interaction and external force. This model was introduced by Malyshev [Probl. Inf. Transm. 51 (2015) 31–36] to study the flow of charged particles on a rigorous mathematical level. It is a simplified version of a 3-dimensional classical Coulomb gas model. We study Gibbs distribution at finite positive temperature extending recent results on the zero temperature case (ground states). We derive the asymptotics for the mean and for the variances of the distances between the neighbouring charges. We prove that depending on the strength of the external...
Louart, Cosme; Liao, Zhenyu; Couillet, Romain
This article studies the Gram random matrix model $G=\frac{1}{T}\Sigma^{{\mathsf{T}}}\Sigma$, $\Sigma=\sigma(WX)$, classically found in the analysis of random feature maps and random neural networks, where $X=[x_{1},\ldots,x_{T}]\in\mathbb{R}^{p\times T}$ is a (data) matrix of bounded norm, $W\in\mathbb{R}^{n\times p}$ is a matrix of independent zero-mean unit variance entries and $\sigma:\mathbb{R}\to\mathbb{R}$ is a Lipschitz continuous (activation) function—$\sigma(WX)$ being understood entry-wise. By means of a key concentration of measure lemma arising from nonasymptotic random matrix arguments, we prove that, as $n,p,T$ grow large at the same rate, the resolvent $Q=(G+\gamma I_{T})^{-1}$, for $\gamma>0$, has a similar behavior as that met in sample covariance matrix models, involving...