Transience/Recurrence and Central Limit Theorem Behavior for Diffusions in Random Temporal Environments

Abstract

Let $\sigma(t)$ be an ergodic Markov chain on a finite state space $E$ and for each $\sigma \in E$, define on $\mathbb{R}^d$ the second-order elliptic operator $L_\sigma = \frac{1}{2} \sum^d_{i,j = 1} a_{ij}(x; \sigma)\frac{\partial^2}{\partial x_i\partial x_j} + \sum^d_{i = 1} b_i(x;\sigma)\frac{\partial}{\partial x_i}.$ Then for each realization $\sigma(t) = \sigma(t, \omega)$ of the Markov chain, $L_{\sigma(t)}$ may be thought of as a time-inhomogeneous diffusion generator. We call such a process a diffusion in a random temporal environment or simply a random diffusion. We study the transience and recurrence properties and the central limit theorem properties for a class of random diffusions. We also give applications to the stabilization and homogenization of the Cauchy problem for random parabolic operators.