Quasistable gradient and hamiltonian systems with a pairwiseinteraction randomly perturbed by wiener processes

[摘要] Infinite systems of stochastic differential equations for randomly perturbed particle systems inRdwith pairwise interacting are considered. For gradient systems these equations are of the formdxk(t)=Fk(t)td+σdwk(t)and for Hamiltonian systems these equations are of the formdx˙k(t)=Fk(t)td+σdwk(t).Herexk(t)is the position of thekth particle,x˙k(t)is its velocity,Fk=−∑j≠kUx(xk(t)−xj(t)),where the functionU:Rd→Ris the potential of the system,σ>0is a constant,{wk(t),k=1,2,…}is a sequence of independent standard Wiener processes.Let{xk}be a sequence of different points inRdwith|xk|→∞, and{υk}be a sequence inRd. Let{x˜kN(t),k≤N}be the trajectories of theN-particles gradient system for whichx˜kN(0)=xk,k≤N, and let{xk(t),k≤N}be the trajectories of theN-particles Hamiltonian system for whichxkN(0)=xk,x˙k(0)=υk,k≤N. A system is called quasistable if for all integersmthe joint distribution of{xkN(t),k≤m}or{x˜kN(t),k≤m}has a limit asN→∞. We investigate conditions on the potential function and on the initial conditions under which a system possesses this property.

[学科分类] 应用数学