Journal of Applied Mathematics and Stochastic Analysis
Volume 16 (2003), Issue 1, Pages 45-67
doi:10.1155/S1048953303000042
Quasistable gradient and hamiltonian systems with a pairwise
interaction randomly perturbed by wiener processes
1Institute of Mathematics, Kiev, Ukraine
2Michigan State University, Department of Statistics and Probability, East Lansing 48824, MI, USA
Received 1 February 2002; Revised 1 May 2002
Copyright © 2003 Anatoli V. Skorokhod. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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.