Infinite systems of stochastic differential equations for randomly perturbed particle systems in Rd with pairwise interacting are considered. For gradient systems these equations are of the form dxk(t)=Fk(t)td+σdwk(t) and for Hamiltonian systems these equations are of the form dx˙k(t)=Fk(t)td+σdwk(t). Here xk(t) is the position of the kth particle, x˙k(t) is its velocity, Fk=−∑j≠kUx(xk(t)−xj(t)), where the function U:Rd→R is the potential of the system, σ>0 is a constant, {wk(t),k=1,2,…} is a sequence of independent standard Wiener processes.

Let {xk} be a sequence of different points in Rd with |xk|→∞, and {υk} be a sequence in Rd. Let {x˜kN(t),k≤N} be the trajectories of the N-particles gradient system for which x˜kN(0)=xk,k≤N, and let {xk(t),k≤N} be the trajectories of the N-particles Hamiltonian system for which xkN(0)=xk,x˙k(0)=υk,k≤N. A system is called quasistable if for all integers m the joint distribution of {xkN(t),k≤m} or {x˜kN(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.