Self-interacting diffusions.
type: Item
instance of: scholarly article
title: Self-interacting diffusions.
zbMATH Open document ID: 1042.60060
author: Michel Benaïm
author: Olivier Raimond
author: Michel Ledoux
publication date: 2003-05-12T00:00:00Z
review text: The authors are concerned with a general class of self-interacting diffusion processes. These are continuous time stochastic processes living on a compact Riemannian manifold \(M\) which can be typically described as solutions to a stochastic differential equation of the form \[ dX_t=dW_t(X_t)-{1\over t} \left(\int_0^t \triangledown V_{X_s} (X_t)\,d s\right)d t \] where \({W_t}\) is a Brownian vector field on \(M\) and \(V_u (x)\) a ``potential'' function. Here, the drift term depends on the normalized occupation measure \(\mu_t = {1\over t} \int_0^t \delta_{X_s} d s\) of the process. The main goal of the paper is to give a systematic treatment of this class of processes and to describe with a great deal of generality the asymptotic behaviour of \(\mu_t\) as \(t\rightarrow\infty\). It is proved that the asymptotic behaviour of \(\mu_t\) can be precisely related to the asymptotic behaviour of a deterministic dynamical semi-flow \(\Phi={\{\Phi_t\}}_{t\geq0}\) defined on the space of the Borel probability measures on \(M\). In particular, the limit sets of \(\mu_t\) are proved to be almost surely attractor free sets for \(\Phi\). These results are applied to several examples of self-attracting/repelling diffusions on the \(n\)-sphere. For instance, in case of self-attracting diffusions, \(\mu_t\) can either converge towards the normalized Riemannian measure, or to a Gaussian measure, depending on the value of the parameter measuring the strength of the attraction. A numerical study is also included.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 1885537
zbMATH Keywords: diffusion
zbMATH Keywords: self-interacting processes
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: occupation measure
reviewed by: Martin Janžura
MaRDI profile type: Publication
full work available at URL: s004400100161
OpenAlex ID: W2070870909
DOI: 10.1007/S004400100161
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1885537