Quasistable gradient and Hamiltonian systems with a pairwise interaction randomly perturbed by Wiener processes
type: Item
instance of: scholarly article
title: Quasistable gradient and Hamiltonian systems with a pairwise interaction randomly perturbed by Wiener processes
zbMATH Open document ID: 1027.60061
DOI: 10.1155/S1048953303000042
author: Anatolii V. Skorokhod
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 2004-01-03T00:00:00Z
full work available at URL: https://eudml.org/doc/50459
review text: 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.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 37Jxx
Mathematics Subject Classification ID: 60G46
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1932174
zbMATH Keywords: gradient and Hamiltonian systems
zbMATH Keywords: Wiener process
zbMATH Keywords: quasistability
reviewed by: Yuliya S. Mishura
MaRDI profile type: Publication
OpenAlex ID: W2147303425
description: scientific article; zbMATH DE number 1932174