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 with pairwise interaction 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 . We investigate conditions on the potential function and on the initial conditions under which a system possesses this property.