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Quasistable gradient and Hamiltonian systems with a pairwise interaction randomly perturbed by Wiener processes. (English) Zbl 1097.60051
Ukrainian mathematical congress – 2001, Kiev, Ukraine, August 21–23, 2001. Proceedings. Section 9. Probability theory and mathematical statistics. Kyïv: Instytut Matematyky NAN Ukraïny (ISBN 966-02-2616-0). 149-159 (2002).

Summary: Infinite systems of stochastic differential equations for randomly perturbed particle systems in ${ℝ}^{d}$ with pairwise interaction are considered. For gradient systems these equations are of the form

$d{x}_{k}\left(t\right)={F}_{k}\left(t\right)dt+\sigma d{w}_{k}\left(t\right)$

and for Hamiltonian systems these equations are of the form

$d{\stackrel{˙}{x}}_{k}\left(t\right)={F}_{k}\left(t\right)dt+\sigma d{w}_{k}\left(t\right)·$

Here ${x}_{k}\left(t\right)$ is the position of the $k$th particle, ${\stackrel{˙}{x}}_{k}\left(t\right)$ is its velocity,

${F}_{k}=-\sum _{j\ne k}{U}_{x}\left({x}_{k}\left(t\right)-{x}_{j}\left(t\right)\right),$

where the function $U:{ℝ}^{d}\to ℝ$ is the potential of the system, $\sigma >0$ is a constant, $\left\{{w}_{k}\left(t\right),k=1,2,\cdots \right\}$ is a sequence of independent standard Wiener processes.

Let $\left\{{x}_{k}\right\}$ be a sequence of different points in ${ℝ}^{d}$ with $|{x}_{k}|\to \infty$, and $\left\{{v}_{k}\right\}$ be a sequence in ${ℝ}^{d}$. Let $\left\{{\stackrel{˜}{x}}_{k}^{N}\left(t\right),k\le N\right\}$ be the trajectories of the $N$-particles gradient system for which ${\stackrel{˜}{x}}_{k}^{N}\left(0\right)={x}_{k},k\le N$, and let $\left\{{x}_{k}\left(t\right),k\le N\right\}$ be the trajectories of the $N$-particles Hamiltonian system for which ${x}_{k}^{N}\left(0\right)={x}_{k},{\stackrel{˙}{x}}_{k}\left(0\right)={v}_{k},k\le N$. A system is called quasistable if for all integers $m$ the joint distribution of $\left\{{x}_{k}^{N}\left(t\right),k\le m\right\}$ or $\left\{{\stackrel{˜}{x}}_{k}^{N}\left(t\right),k\le m\right\}$ has a limit as $N\to \infty$. We investigate conditions on the potential function and on the initial conditions under which a system possesses this property.

##### MSC:
 60H10 Stochastic ordinary differential equations 60G46 Martingales and classical analysis 60K35 Interacting random processes; statistical mechanics type models; percolation theory