Large deviations for empirical process of mean-field interacting particle system with unbounded jumps
type: Item
instance of: scholarly article
title: Large deviations for empirical process of mean-field interacting particle system with unbounded jumps
zbMATH Open document ID: 0852.60029
author: Shui Feng
published in: The Annals of Probability
publication date: 1996-12-02T00:00:00Z
review text: A particle system with mean-field interaction is considered. It is described as an \(N\)-dimensional Markov process. It is known that, for any finite time interval, the corresponding empirical process satisfies a law of large number as \(N \to \infty\) and that the limiting deterministic dynamics are characterized as a weak solution of a nonlinear master equation. The nonlinear master equation was proposed by \textit{G. Nicolis} and \textit{I. Prigogine} [``Self-organization in nonequilibrium systems. From dissipative structures to order through fluctuations'' (1977; Zbl 0363.93005)]. \textit{S. Feng} and \textit{X. Zheng} [Stochastic Processes Appl. 43, No. 1, 65-84 (1992; Zbl 0762.60075)] proved the existence and uniqueness of the solution to the nonlinear master equation. For the second Schögl model, the existence of at least three equilibrium states is proved. The motivation of the paper is to investigate the long-time behavior such as tunnelling and metastability of the \(N\)-particle systems. Consider the case when the nonlinear master equation has more than one equilibrium state. From the law of large numbers we can see that if \(N\) is large but finite, then the empirical process will normally follow the path of the dynamical system underlying the nonlinear master equation, that is attracted by one of the equilibrium states. As time goes on, the particle system will make small fluctuations near the equilibrium states. However, because of ergodicity, the transition from the neighborhood of one equilibrium state to another may eventually occur via large deviations. The purpose of the paper is to study the large deviations of the empirical process and to find a nice representation for the action functional. The special feature of the model discussed in the paper is that it is not only a pure jump Markov process with mean-field interaction, but also with unbounded jumps. Similar problems have been discussed by \textit{M. I. Frejdlin} and \textit{A. D. Venttsel'}…
reviewed by: Birutė Kryžienė
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82C26
zbMATH DE Number: 828378
zbMATH Keywords: \(Q\)-systems
zbMATH Keywords: pure jump Markov process
zbMATH Keywords: nonlinear master equation
zbMATH Keywords: Schlögl model
zbMATH Keywords: meanfield interacting particle system
zbMATH Keywords: empirical process
zbMATH Keywords: large deviation system
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988496
OpenAlex ID: W1995573595
DOI: 10.1214/AOP/1176988496
description: scientific article; zbMATH DE number 828378