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 Z d , where T is the circle, and defined by an infinitesimal generator L which acts on local functions f as Lf(η)=∑ i∈ Z d ( a i 2(η i) 2 ∂ 2f ∂η i 2 +b i(η) ∂f ∂η i ) . 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 η i . Suppose that there is a product measure ν which is invariant. Then if ν is the Lebesgue measure or if d=1,2, it is the unique invariant measure. Furthermore, if ν 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} Z d , with uniformly positive bounded flip rates which are finite range. To cite this article: A.F. Ramı́rez, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 139–144