Gradient dynamics of infinite point systems
type: Item
instance of: scholarly article
title: Gradient dynamics of infinite point systems
zbMATH Open document ID: 0623.60119
author: B. George
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: A countable set S of d-dimensional particles suspended in a fluid is considered, where the interaction is given by a pair potential \(U: {\mathbb{R}}^ d\to (-\infty,\infty]\) with finite range. The model is represented by the infinite system of stochastic differential equations \[ d\omega_ k=-\sum_{j\neq k}\text{grad} U(\omega_ k-\omega_ j)dt+\sigma dw_ k,\quad k\in S, \] where \((\omega_ k)_{k\in S}\) is the configuration of the particle system, \((w_ k)_{k\in S}\) is a family of independent standard Wiener processes in \({\mathbb{R}}^ d\), and \(\sigma\geq 0\) is a constant. The purpose of the paper is to study existence and regularity properties of solutions of this system of equations. For smooth interactions, existence and uniqueness of nonequilibrium solutions is proved if \(d\leq 4\), and for a singular U if \(d\leq 2\) and \(\sigma =0\). Solutions are constructed as a.s. limits of solutions of finite subsystems. Properties of the Markov semigroups are investigated in order to develop prerequisites for the theory of hydrodynamical fluctuations in equilibrium.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 4011629
zbMATH Keywords: interacting Brownian particles
zbMATH Keywords: infinite system of stochastic differential equations
zbMATH Keywords: smooth interactions
zbMATH Keywords: existence and uniqueness of nonequilibrium solutions
zbMATH Keywords: Markov semigroups
zbMATH Keywords: hydrodynamical fluctuations in equilibrium
reviewed by: Luis G. Gorostiza
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992156
OpenAlex ID: W2057881547
DOI: 10.1214/AOP/1176992156
description: scientific article; zbMATH DE number 4011629