Gradient dynamics of infinite point systems. (English) Zbl 0623.60119

Fritz, J.

Ann. Probab. 15, 478-514 (1987).

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.

Reviewer: L.G.Gorostiza

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J35	Transition functions, generators and resolvents

Keywords:

interacting Brownian particles; infinite system of stochastic differential equations; smooth interactions; existence and uniqueness of nonequilibrium solutions; Markov semigroups; hydrodynamical fluctuations in equilibrium