Quasistable gradient and Hamiltonian systems with a pairwise interaction randomly perturbed by Wiener processes
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.
scientific article; zbMATH DE number 1932174