An elementary proof of the uniqueness of invariant product measures for some infinite dimensional processes

Abstract

Consider an infinite dimensional diffusion process with state space T-Zd, where T is the circle, and defined by an infinitesimal generator L which acts on local functions f as Lf(eta) = Sigma(iepsilonZ)(d) (a(i)(2)(etai))/2 partial derivative(2) f/partial derivative(etai)(2) + b(i)(eta) partial derivativef/partial derivative(etai). Suppose that the coefficients a(i) and b(i) are smooth, bounded, of finite range, have uniformly bounded second order partial derivatives, that a(i) are uniformly bounded from below by some strictly positive constant, and that a(i) is a function only of eta(i). Suppose that there is a product measure v which is invariant. Then if v is the Lebesgue measure or if d = 1, 2, it is the unique invariant measure. Furthermore, if v is translation invariant, it is the unique invariant, translation invariant measure. The proofs are elementary. Similar results can be proved in the context of an interacting particle system with state space {0, 1}(Zd) with uniformly positive bounded flip rates which are finite range. (C) 2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.