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

Let (t) be an ergodic Markov chain on a finite state space E and for each E, define on Rᵈ the second-order elliptic operator L_ = 12 ᵈ₈, ₉ = ₁ a₈₉ (x;) ² xᵢ xⱼ + ᵈ₈ = ₁ bᵢ (x;) xᵢ. Then for each realization (t) = (t, ) of the Markov chain, L (ₓ) 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.