Nonlinear self-stabilizing processes. II: Convergence to invariant probability
type: Item
instance of: scholarly article
title: Nonlinear self-stabilizing processes. II: Convergence to invariant probability
zbMATH Open document ID: 0932.60064
DOI: 10.1016/S0304-4149(98)00019-2
author: Bernard Roynette
author: Said Benachour
author: Pierre Vallois
published in: Stochastic Processes and their Applications
publication date: 1999-11-18T00:00:00Z
review text: [For part I see above.] Let us consider nonlinear stochastic differential equations \[ X_{t} = X_{0} + B_{t} - \frac 12\int ^{t}_{0} b(s, X_{s}) \roman ds, \quad b(s,x) = \mathbf E\beta (x-X_{s}),\tag{1} \] where \(B\) is a standard Brownian motion and \(\beta :\mathbb{R}\to \mathbb{R}\) is an odd, increasing, locally Lipschitz continuous function of a polynomial growth, convex on \(\mathbb{R}_{+}\). In the first part of the paper the authors proved that there exists a unique invariant density \(u\) for the equation (1) if \(\beta (x)=\beta_{0}(x)+\alpha x\), where \(\alpha >0\) is large enough, and \(\beta_{0}\) is an odd, increasing, locally Lipschitz function with a polynomial growth, satisfying \(|\beta_{0} (x)|\geq k|x|^\varrho \) for some \(\varrho >1\) and any \(|x|\geq 1\). In the second part the weak convergence of the solution \(X_{t}\) to the invariant probability \(u(x) dx\) as \(t\to \infty \) is established under the above hypotheses.
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1364479
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: invariant density
zbMATH Keywords: convergence in distribution
MaRDI profile type: Publication
cites work: Q5186515
cites work: Nonlinear self-stabilizing processes. I: Existence, invariant probability, propagation of chaos
cites work: Q5336643
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cites work: Some remarks on ultracontractivity
Wikidata QID: Q127109659
description: scientific article; zbMATH DE number 1364479