Relative entropy and mixing properties of infinite dimensional diffusions

Abstract

Let η be a diffusion process taking values on the infinite dimensional space T Z , where T is the circle, and with components satisfying the equations dη i =σ i (η) dW i +b i (η) dt for some coefficients σ i and b i , i∈Z. Suppose we have an initial distribution μ and a sequence of times t n →∞ such that lim n →∞μS tn =ν exists, where S t is the semi-group of the process. We prove that if σ i and b i are bounded, of finite range, have uniformly bounded second order partial derivatives, and inf i ,ησ i (η)>0, then ν is invariant.