Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type
type: Item
instance of: scholarly article
title: Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type
zbMATH Open document ID: 1189.60123
author: Thomas G. Kurtz
author: Peter M. Kotelenez
publication date: 2010-01-15T00:00:00Z
review text: Under appropriate Lipschitz conditions, \textit{P. Kotelenez} [Probab. Theory Relat. Fields 102, 159--188 (1995; Zbl 0821.60066)] proved the strong Itô existence and uniqueness of solutions for the system of stochastic ordinary differential equations (SODEs) \[ dr^i(t)=F_{\varepsilon}(r^i(t),X(t),t)dt+\int {\mathcal J}_{\varepsilon}(r^i(t),q,X(t),t)w(dq,dt) \] where \(w_l(dq,dt)\) are i.i.d. Gaussian standard white noise random fields on \({\mathbb R}^d \times {\mathbb R}_+, l=1,\ldots,d\), \(X(t)\) is the empirical distribution of the \(r^i(t), i=1,\dots,N\), and \(\varepsilon >0\) is a correlation parameter. The empirical distributions \(X(t)\) are solutions of quasilinear stochastic partial differential equations (SPDEs). After passing to a continuum limit, the solutions of the SPDEs are proved to converge to the solution of a deterministic quasilinear partial differential equation as the correlation parameter \(\varepsilon\) tends to \(0\). The limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes. This approach is compared with the one of Kunita about SODEs driven by a Gaussian space-time random field.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60K40
Mathematics Subject Classification ID: 60G09
zbMATH DE Number: 5659494
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: partial differential equations
zbMATH Keywords: macroscopic limit
zbMATH Keywords: particle systems
zbMATH Keywords: stochastic ordinary differential equations
zbMATH Keywords: exchangeable sequences
reviewed by: Dominique Lépingle
MaRDI profile type: Publication
full work available at URL: s00440-008-0188-0
OpenAlex ID: W2043295228
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DOI: 10.1007/S00440-008-0188-0
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 5659494