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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 d with pairwise interaction are considered. For gradient systems these equations are of the form

dx k (t)=F k (t)dt+σdw k (t)

and for Hamiltonian systems these equations are of the form

dx ˙ k (t)=F k (t)dt+σdw k (t)·

Here x k (t) is the position of the kth particle, x ˙ k (t) is its velocity,

F k =- jk U x (x k (t)-x j (t)),

where the function U: d is the potential of the system, σ>0 is a constant, {w k (t),k=1,2,} is a sequence of independent standard Wiener processes.

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

60H10Stochastic ordinary differential equations
60G46Martingales and classical analysis
60K35Interacting random processes; statistical mechanics type models; percolation theory