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Quasistable gradient and Hamiltonian systems with a pairwise interaction randomly perturbed by Wiener processes. (English) Zbl 1027.60061
Let $$w_k$$ be a sequence of independent standard Wiener processes in $$R^d$$, $$x_k$$ be a finite or infinite sequence of $$R^d$$-valued stochastic processes that describe the evolution of a set of randomly perturbed pairwise interacting particles. Gradient and Hamiltonian systems of the form $dx_k(t)=F_k(t) dt+\sigma dw_k(t) \quad\text{and}\quad d\dot{x}_k(t)=F_k(t) dt+ \sigma dw_k(t),$ respectively, are considered, where $$F_k=-\sum_{j\neq k}U_k(x_k(t)-x_j(t))$$. The assumptions on the potential $$U$$ and on initial conditions, sufficient for the quasistability of both the systems, are established. Here quasistability means that any joint distributions of corresponding finite $$N$$-particle systems have a limit as $$N\rightarrow\infty$$. The main assumptions include the existence of gradient (Hamiltonian) free particle system, the relation $$\lim_{s\downarrow 0}s^{\alpha+1}u''(s)=a$$ for $$\alpha<1$$, and some estimates for the initial configuration.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H30 Applications of stochastic analysis (to PDEs, etc.) 34F05 Ordinary differential equations and systems with randomness 37Jxx Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 60G46 Martingales and classical analysis 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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