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 R d with pairwise interacting are considered. For gradient systems these equations are of the form d x k ( t ) = F k ( t ) t d + σ d w k ( t ) and for Hamiltonian systems these equations are of the form d x ˙ k ( t ) = F k ( t ) t d + σ d w k ( t ) . Here x k ( t ) is the position of the k th particle, x ˙ k ( t ) is its velocity, F k = − ∑ j ≠ k U x ( x k ( t ) − x j ( t ) ) , where the function U : R d → R 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 R d with | x k | → ∞ , and { υ k } be a sequence in R d . Let { x ˜ k N ( t ) , k ≤ N } be the trajectories of the N -particles gradient system for which x ˜ k N ( 0 ) = x k , k ≤ N , and let { x k ( t ) , k ≤ N } be the trajectories of the N -particles Hamiltonian system for which x k N ( 0 ) = x k , x ˙ k ( 0 ) = υ k , k ≤ N . A system is called quasistable if for all integers m the joint distribution of { x k N ( t ) , k ≤ m } or { x ˜ k N ( t ) , k ≤ m } has a limit as N → ∞ . We investigate conditions on the potential function and on the initial conditions under which a system possesses this property.