QUASISTABLE GRADIENT AND HAMILTONIAN SYSTEMS WITH A PAIRWISE INTERACTION RANDOMLY PERTURBED BY WIENER PROCESSES

Infinite systems of stochastic differential equations for randomly perturbed particle systems in (cid:1) (cid:2) with pairwise interacting are considered. For gradient systems these equations are of the form and for Hamiltonian systems these equations are of the form (cid:1) Here is the position of the th particle, is its velocity, , and let be the trajectories of the -particles Hamiltonian system for which , , . A system is called quasistable if for all (cid:29) . We investigate conditions on the potential function and on the initial conditions under which a system possesses this property.

Infinite systems of stochastic differential equations for randomly perturbed particle systems in with pairwise interacting are considered. For gradient systems these equations are of the form and for Hamiltonian systems these equations are of the form Here is the position of the th particle, is its velocity, where the function is the potential of the system, is a constant, is a sequence of independent standard Wiener processes. Let be a sequence of different points in with , and be a sequence in . Let be the trajectories of the -particles gradient system for which ~ , and let be the trajectories of the -particles Hamiltonian system for which , , . A system is called quasistable if for all integers the joint distribution of or has a limit as 1 Introduction We consider a finite or an infinite sequence of -valued stochastic processes " where is a finite or infinite interval of the natural numbers by which the evolution of a set of pairwise interacting particles in the space is described. The interaction is determined by a potential satisfying the condition ( ) where the function is smooth, for P # # # $ % & , and for some the relation is fulfilled, , are constants. % ( The motion of a gradient system is determined by the system of the stochastic differential equations The number of equations coincides with the number of particles.
is the sequence of independent standard Wiener processes in . These equations are solved with some initial conditions , where , is a sequence of different points in " " , for which lim if is an infinite interval. The motion of a Hamiltonian system is determined by the system of the stochastic differential equations Here is the velocity of the th particle. Equations (2) are solved with the  initial conditions   ,   where is another sequence in . " The existence of the solution to infinite systems of kind (1) was considered for smooth functions by J. Fritz (see [2]). Finite gradient and Hamiltonian systems with the potentials satisfying condition ( ) were considered by the author (see [5]). P . Let sequences and be given. For any Quasistability * * , there exists the solution to system (1) , satisfying the initial conditions and the solution to system (2) satisfying the initial conditions + ~~ This follows from the results of [5].
Gradient system (1) is called quasistable if for any the joint distribution of the ! stochastic processes (t), has as a limit in the space . ! , ! - (2) is called quasistable if for any the joint distributions of the ! stochastic processes has a limit in the same space. Note that thẽ ! quasistability is determined by the potential and initial conditions.

Hamiltonian system
The main goal of this article is the investigation of the conditions on the potential and initial conditions of gradient and Hamiltonian systems under which they are quasistable.

The Spaces and
It is convenient to consider gradient systems (1)  and for a continuous function with bounded support Using Ito's formula and considering the function as a function of two variables: '), ( ( we can rewrite system (1) using -valued function for which in the form where " + % 0 )) and is the set of those for which ' and '' are continuous functions. 0 0 " A -valued stochastic process is called a weak solution to equation (4) if for all " 0 the stochastic process is a martingale with respect to the filtration with the square characteristic If is a weak solution to system (4) and with the distance is a separable locally compact space. It is convenient to consider Hamiltonian systems in the configuration space which is the subspace of those locally finite counting measures in for which ; where are sequences in the space . Let be a solution to equation (2). Introduce a -valued function Let be a function from . Ito's formula implies the relations ; is a continuous martingale with the square characteristic 3 Some Properties of Solutions to Finite Systems 50 A. SKOROHOD Assume that and the function satisfies the condition ( ). In this section we P consider solutions to systems (1) and (2)  inf . Further proof of statement b) is the same as for statement a). The lemma is proved.

Free Particles Processes
We consider now the solution to equations (1) and (2) for . They are called free particles processes. The stochastic process in the space ( is called an -particles free gradient process, and the stochastic process The measure is absolutely continuous with respect to the measurẽ ! / ! ! ! , and the measure is absolutely continuous with respect to the measure for all Proof: Denote by the distribution of the stochastic process ! ; / D / ~~ / in the space . This stochastic process was introduced in the proof of Lemma 1.

-5 /
It follows from the results of [5] that where is calculated by formula (22)  Here is the expectation with respect to the joint distribution of the Wiener processes K E 9 " D "

O
We will use some graphs in the set O Denote by the set of all connected graphs with the set of vertices Q R Q D , and the set of edges The set of all minimal graphs is denoted by . T Q Lemma 6: There exist some constants , , and a graph for which the inequality is valid It is easy to see that the inequality is valid It is easy to see that the sets of r.v. are independent for different as well as the sets . Using the Cauchy inequality we can write where the sets are determined by the sets which are determined by their properties; with some . This formula and formulas (37),(38) imply the proof of the lemma.

The Condition of the Quasistability of Gradient Systems
Consider the graph introduced in the proof of Lemma 6. Let  )) ) )) the relation is fulfilled for some . For , we set Q Q " Q " T Z Q C 4?
)) ) )) Introduce the functions so we will write K Q K Q 49 9 4 ~~ ) ) the function in the right-hand side of the equality depends only of those for which 9 4 O O O " Q Q 9 $ ) ) ) ) 9