Ergodicity for infinite particle systems with locally conserved quantities
type: Item
instance of: scholarly article
zbMATH Open document ID: 1268.60120
DOI: 10.1142/S0219025712500051
published in: Infinite Dimensional Analysis, Quantum Probability and Related Topics
publication date: 2012-06-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1002.0282
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 82C22
zbMATH DE Number: 6044893
zbMATH Keywords: Hörmander-type generator
zbMATH Keywords: locally conserved quantity
zbMATH Keywords: Ligget-Nash inequality
MaRDI profile type: Publication
arXiv ID: 1002.0282
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OpenAlex ID: W2963989028
title: Ergodicity for infinite particle systems with locally conserved quantities
author: J. Inglis
author: M. Neklyudov
author: Boguslaw Zegarliński
review text: The paper analyses degenerate infinite-dimensional sub-elliptic generators, and obtains estimates on the long-term behavior of the corresponding Markov semigroups that describe a model of heat conduction. The authors also establish ergodicity of the system for a family of invariant measures, and show that the optimal rate of convergence to the equilibrium state is polynomial. Also, the Liggett-Nash-type inequality appears to be valid. Section 2 introduces the basic notation, and states an infinite system of stochastic differential equations of interest. The existence of a mild solution is considered in Section 3, whereas Section 4 discusses general properties of the corresponding semigroup, in particular, the existence of a family of invariant measures, strong continuity, positivity, and contractivity properties in \(L^2\)-spaces. Section 5 provides a certain characterisation of invariant sub-spaces (of Sobolev type). Ergodicity with optimal rate of convergence to the equilibrium state is demonstrated in Section 6. Section 7 derives the Liggett-Nash-type inequalities on the basis of the obtained results. A generalized dynamics of the related type, complemented with some extra features, is the topic of Section 8. It deals with the phase transitions in the stochastic paradigm. The paper also shows that, in the considered families, one can observe a change in the behavior of the decay to the equilibrium state from exponential to algebraic. Further application of the ergodicity results are in the focus of Section 9. The paper includes the detailed proofs of the aforementioned results.
reviewed by: Eugen Mamontov
description: scientific article; zbMATH DE number 6044893