Journal of Theoretical Probability最新文献

Let (X={ X(t), tin mathbb {R}^{N}} ) be a centered space-time anisotropic Gaussian random field in (mathbb {R}^d) with stationary increments, where the components (X_{i}(i=1,ldots ,d)) are independent but distributed differently. Under certain conditions, we not only give the Hausdorff dimension of the graph sets of X in the asymmetric metric in the recurrent case, but also determine the exact Hausdorff measure functions of the graph sets of X in the transient and recurrent cases, respectively. Moreover, we establish a uniform Hausdorff dimension result for the image sets of X. Our results extend the corresponding results on fractional Brownian motion and space or time anisotropic Gaussian random fields.

We introduce the voter model on the infinite integer lattice with a slow membrane and investigate its hydrodynamic behavior and nonequilibrium fluctuations. The voter model is one of the classical interacting particle systems with state space ({0,1}^{mathbb Z^d}). In our model, a voter adopts one of its neighbors’ opinion at rate one except for neighbors crossing the hyperplane ({x:x_1 = 1/2}), where the rate is (alpha N^{-beta }) and thus is called a slow membrane. Above, (alpha >0 textrm{and} beta ge 0) are given parameters and the positive integer N is a scaling parameter. We consider the limit (N rightarrow infty ) and prove that the hydrodynamic limits are given by the heat equation without or with Robin/Neumann conditions depending on the values of (beta ). We also consider the nonequilibrium fluctuations, where the limit is described by generalized Ornstein–Uhlenbeck processes with certain boundary conditions corresponding to the hydrodynamic equation.

We consider rough differential equations whose coefficients contain path-dependent bounded variation terms and prove the existence and a priori estimate of solutions. These equations include classical path-dependent stochastic differential equations containing running maximum processes and normal reflection terms. We apply these results to determine the topological support of the solution processes.

Abstract

We study the homogenization problem for a system of stochastic differential equations with local time terms that models a multivariate diffusion in the presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular interface-induced drift.

Let (eta _1), (eta _2,ldots ) be independent copies of a random variable (eta ) with zero mean and finite variance which is bounded from the right, that is, (eta le b) almost surely for some (b>0). Considering different types of the asymptotic behaviour of the probability (mathbb {P}{eta in [b-x,b]}) as (xrightarrow 0+), we derive precise tail asymptotics of the random Dirichlet series (sum _{kge 1}k^{-alpha }eta _k) for (alpha in (1/2, 1]).

The problem of evaluating the accuracy of Poisson approximation to the distribution of a sum of independent integer-valued random variables has attracted a lot of attention in the past six decades. Among authors who contributed to the topic are Prokhorov, Kolmogorov, LeCam, Shorgin, Barbour, Hall, Deheuvels, Pfeifer, Roos, and many others. From a practical point of view, the problem has important applications in insurance, reliability theory, extreme value theory, etc. From a theoretical point of view, the topic provides insights into Kolmogorov’s problem concerning the accuracy of approximation of the distribution of a sum of independent random variables by infinitely divisible laws. The task of establishing an estimate of the accuracy of Poisson approximation with a correct (the best possible) constant at the leading term remained open for decades. We present a solution to that problem in the case where the accuracy of approximation is evaluated in terms of the point metric. We generalise and sharpen the corresponding inequalities established by preceding authors. A new result is established for the intensively studied topic of compound Poisson approximation to the distribution of a sum of integer-valued r.v.s.

Gaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the n-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below).

In this paper, we establish moment and Bernstein-type inequalities for additive functionals of geometrically ergodic Markov chains. These inequalities extend the corresponding inequalities for independent random variables. Our conditions cover Markov chains converging geometrically to the stationary distribution either in weighted total variation norm or in weighted Wasserstein distances. Our inequalities apply to unbounded functions and depend explicitly on constants appearing in the conditions that we consider.

This paper deals with the long-term behavior of the solution to the nonlinear stochastic heat equation (frac{partial u}{partial t} - frac{1}{2}Delta u = b(u){dot{W}}), where b is assumed to be a globally Lipschitz continuous function and the noise ({dot{W}}) is a centered and spatially homogeneous Gaussian noise that is white in time. We identify a set of nearly optimal conditions on the initial data, the correlation measure of the noise, and the weight function (rho ), which together guarantee the existence of an invariant measure in the weighted space (L^2_rho ({mathbb {R}}^d)). In particular, our result covers the parabolic Anderson model (i.e., the case when (b(u) = lambda u)) starting from the Dirac delta measure.

We consider Bernoulli bond and site percolation on Cayley graphs of reflection groups in the three-dimensional hyperbolic space ({mathbb {H}^3}) corresponding to a very large class of Coxeter polyhedra. In such setting, we prove the existence of a non-empty non-uniqueness percolation phase, i.e. that (p_c < p_u). This means that for some values of the Bernoulli percolation parameter there are a.s. infinitely many infinite components in the percolation subgraph. The proof relies on upper estimates for the spectral radius of the graph and on a lower estimate for its growth rate. The latter estimate involves only the number of generators of the group and is proved in the article as well.