An elementary proof of the uniqueness of invariant product measures for some infinite dimensional processes
type: Item
instance of: scholarly article
title: An elementary proof of the uniqueness of invariant product measures for some infinite dimensional processes
zbMATH Open document ID: 0998.60093
DOI: 10.1016/S1631-073X(02)02201-X
author: Alejandro F. Ramírez
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2002-11-25T00:00:00Z
review text: Let \(T\) denote the unit circle. Consider a finite range elliptic infinite-dimensional diffusion on the torus \(T^{\mathbb{Z}^d}\), \(d\geq 1\), whose coefficients are bounded and satisfy other regularity conditions. Let \(\nu\) be a product measure which is invariant for the process considered. Then (i) if \(\nu\) is translation invariant, then it is the unique invariant, translation invariant measure; (ii) if \(d=1\) or 2, then \(\nu\) is the unique invariant measure; (iii) if \(\nu\) is Lebesgue measure, then it is the unique invariant measure. These results generalize previous results of the author [Probab. Theory Relat. Fields 110, No.~3, 369-395 (1998; Zbl 0929.60081)]. The proofs are elementary. The results obtained are partially extended to some interacting particle systems with state space \([0,1]^{\mathbb{Z}^d}\).
reviewed by: Marius Iosifescu
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 1744368
zbMATH Keywords: finite range elliptic infinite-dimensional diffusion
zbMATH Keywords: translation invariant measure
zbMATH Keywords: interacting particle systems
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1744368