ESAIM: Probability and Statistics

Table of Contents - 2002 - Volume 6, New directions in Time Series Analysis (Guest Editor: Philippe Soulier)  

Research Article

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

Ramírez, Alejandro F.

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

Abstract

Consider an infinite dimensional diffusion process process on TZd, where T is the circle, defined by the action of its generator L on C2(TZd) 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, ai and bi are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that ai is only a function of $\eta_i$ and that $\inf_{i,\eta}a_i(\eta)>0$. 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 $Lf(\eta)=\sum_{x\in{\bf Z}^d}c(x,\eta)(f(\eta^x)-f(\eta))$, where $\eta^x$ is the configuration obtained from η 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 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.

(Received December 15 2001)

(Online publication November 15 2002)

Key Words:

  • Infinite dimensional diffusions;
  • Malliavin calculus;
  • Interacting particles systems.

Mathematics Subject Classification:

  • 82C20;
  • 82C22;
  • 60H07;
  • 60K35
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