Self-attracting diffusions: Case of the constant interaction
type: Item
instance of: scholarly article
title: Self-attracting diffusions: Case of the constant interaction
zbMATH Open document ID: 0881.60055
author: Olivier Raimond
publication date: 1998-02-03T00:00:00Z
review text: The paper deals with a stochastic equation of the form \[ dX_t= dB_t-\sigma \int^t_0 {X_t-X_s \over |X_t- X_s|} ds dt \] with \(X_0=0\), where \(\sigma\) is a positive constant and \(B_t\) is a \(d\)-dimensional Brownian motion. This equation describes a diffusion process attracted by its own trajectories. It has been studied before by M. Cranston and Y. Le Jan in the one-dimensional case. In the present paper the general \(d\)-dimensional case is considered. The main theorem states that the solution \(X_t\) converges almost surely. The proof requires new ideas with respect to the one-dimensional case, and in particular a good understanding of the action of the restoring drift.
reviewed by: Franco Flandoli
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1002143
zbMATH Keywords: self-attracting diffusions
zbMATH Keywords: diffusions
zbMATH Keywords: restoring drift
zbMATH Keywords: stochastic equation
zbMATH Keywords: Brownian motion
MaRDI profile type: Publication
full work available at URL: s004400050082
OpenAlex ID: W1977076560
DOI: 10.1007/S004400050082
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1002143