Uniqueness of invariant product measures for elliptic infinite dimensional diffusions and particle spin systems

Abstract

Consider an infinite dimensional diffusion process process on T Z d , where T is the circle, defined by the action of its generator L on C2 (T Z d ) local functions as . Assume that the coefficients, a i and b i are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that a i is only a function of and that . Suppose ν is an invariant product measure. Then, if ν is the Lebesgue measure or if d=1,2 , it is the unique invariant measure. Furthermore, if ν is translation invariant, then it is the unique invariant, translation invariant measure. Now, consider an infinite particle spin system, with state space {0,1}Z d , defined by the action of its generator on local functions f by , where is the configuration obtained from η altering only the coordinate at site x . Assume that are of finite range, bounded and that . Then, if ν is an invariant product measure for this process, ν is unique when d=1,2 . Furthermore, if ν is translation invariant, it is the unique invariant, translation invariant measure. The proofs of these results show how elementary methods can give interesting information for general processes.