We investigate the dynamics of an infinite neural network in the case of local inhibitory connections and analyse their large-time limit behaviour. We show that for a certain set of parameters the net is ergodic, and that the convergence to the invariant measure is exponentially fast.

Several useful point processes such as the Markovian arrival process, the input and departure processes of finite-capacity Markovian queues, and various models for counters and biological phenomena are obtained by considering Markov chains with marked transitions. This point of view yields many...

In this paper we deal with the solutions of Itô stochastic differential equationfor a small parameter [epsilon]. We prove that for 0[less-than-or-equals, slant][alpha]<1 and V a divergence-free, Gaussian random field, sufficiently strongly mixing in t variable the family of processes {X[epsilon](t)}t[greater-or-equal, slanted]0, [epsilon]>0 converges weakly to a Brownian motion. The entries of the covariance matrix of the limiting Brownian motion are given by...</1>

We give a probabilistic interpretation of the solution of a diffusion-convection equation. To do so, we define a martingale problem in which the drift coefficient is nonlinear and unbounded for small times whereas the diffusion coefficient is constant. We check that the time marginals of any...

We study different examples of singular perturbations of one-dimensional stochastic differential equations. We derive limit theorems for random perturbations of dynamical systems and diffusions in a random environment.

Let be a real-valued Lévy process and the time spent on before time . Suppose that 0 is not polar. We determine the distribution of where is the first return time to 0 in the irregular case, and the inverse local time at 0 in the regular case. This generalizes a recent result of Fitzsimmons and...

The aim of this paper is to investigate exponential stability of paths for a class of Hilbert space-valued non-linear stochastic evolution equations. The analyses consist in using exponential martingale formula, Lyapunov functional and some special inequalities derived for our stability...

For the problem of estimating the offspring mean of a branching process with immigration, we propose a modification of the sequential estimator of considered in Sriram et al. (, Ann. Statist.) and study its nonasymptotic and asymptotic properties. In the nonasymptotic setting, it is shown that...

This paper is concerned with the approximation of a class of open population epidemic models by time-inhomogeneous birth-and-death processes. In particular, we consider models in which the population of susceptibles behaves in the absence of infection as a general branching process. It is shown...

The range of random walks means the number of distinct sites visited at least once by the random walk before time n. We study an almost sure invariance principle for the range of random walks on the four or more dimensional integer lattice and obtain that the centralized and linearly...