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

Free access article

Issue		ESAIM: PS Volume 6, 2002


Page(s)		147 - 155
DOI		10.1051/ps:2002008

ESAIM: P&S, July 2002, Vol. 6, pp. 147-155
DOI: 10.1051/ps:2002008

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

Alejandro F. Ramírez

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

(Received December 15, 2001.)

Abstract
Consider an infinite dimensional diffusion process process on $T^{{\bf Z}^d}$ , where T is the circle, defined by the action of its generator L on $C^2(T^{{\bf Z}^d})$ local functions as $Lf(\eta)=\sum_{i\in{\bf Z}^d}\left(\frac{1}{2}a_i \frac{\partial^2 f}{\partial \eta_i^2}+b_i\frac{\partial f}{\partial \eta_i}\right)$ . 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 $\eta_i$ and that $\inf_{i,\eta}a_i(\eta)>0$ . Suppose $\nu$ is an invariant product measure. Then, if $\nu$ is the Lebesgue measure or if d=1,2, it is the unique invariant measure. Furthermore, if $\nu$ is translation invariant, then it is the unique invariant, translation invariant measure. Now, consider an infinite particle spin system, with state space $\{0,1\}^{{\bf Z}^d}$ , defined by the action of its generator on local functions f by $Lf(\eta)=\sum_{x\in{\bf Z}^d}c(x,\eta)(f(\eta^x)-f(\eta))$ , where $\eta^x$ is the configuration obtained from $\eta$ altering only the coordinate at site x. Assume that $c(x,\eta)$ are of finite range, bounded and that $\inf_{x,\eta}c(x,\eta)>0$ . Then, if $\nu$ is an invariant product measure for this process, $\nu$ is unique when d=1,2. Furthermore, if $\nu$ 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.

Mathematics Subject Classification. 82C20, 82C22, 60H07, 60K35.

Key words: Infinite dimensional diffusions, Malliavin calculus, Interacting particles systems.

© EDP Sciences, SMAI 2002

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