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

Let σ(t) be an ergodic Markov chain on a finite state space E and for each σ ∈ E, define on Rᵈ the second-order elliptic operator L_σ = 1/2 ∑ᵈi,j = 1 aᵢⱼ(x; σ)∂²/∂ xᵢ∂ xⱼ + ∑ᵈi = 1 bᵢ(x;σ)∂/∂ xᵢ. Then for each realization σ(t) = σ(t, ω) of the Markov chain, Lσ(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.