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

Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile; aramirez@mat.puc.cl.

Received: 15 December 2001

Abstract

Consider an infinite dimensional diffusion process process on T^Zd, where T is the circle, defined by the action of its generator L on C²(T^Zd) 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}^Zd, 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.