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Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type
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  • Published: 12 December 2008

Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type

  • Peter M. Kotelenez1 &
  • Thomas G. Kurtz2 

Probability Theory and Related Fields volume 146, Article number: 189 (2010) Cite this article

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Abstract

A class of quasilinear stochastic partial differential equations (SPDEs), driven by spatially correlated Brownian noise, is shown to become macroscopic (i.e., deterministic), as the length of the correlations tends to 0. The limit is the solution of a quasilinear partial differential equation. The quasilinear SPDEs are obtained as a continuum limit from the empirical distribution of a large number of stochastic ordinary differential equations (SODEs), coupled though a mean-field interaction and driven by correlated Brownian noise. The limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes. We also compare our approach to SODEs with the one introduced by Kunita.

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Authors and Affiliations

  1. Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH, 44106, USA

    Peter M. Kotelenez

  2. Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI, 53706-1388, USA

    Thomas G. Kurtz

Authors
  1. Peter M. Kotelenez
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  2. Thomas G. Kurtz
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Corresponding author

Correspondence to Peter M. Kotelenez.

Additional information

This research was partially supported by NSF grant DMS 05-03983.

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Kotelenez, P.M., Kurtz, T.G. Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type. Probab. Theory Relat. Fields 146, 189 (2010). https://doi.org/10.1007/s00440-008-0188-0

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  • Received: 23 August 2006

  • Revised: 22 October 2008

  • Published: 12 December 2008

  • DOI: https://doi.org/10.1007/s00440-008-0188-0

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Keywords

  • Stochastic partial differential equations
  • Partial differential equations
  • Macroscopic limit
  • Particle systems
  • Stochastic ordinary differential equations
  • Exchangeable sequences

Mathematics Subject Classification (2000)

  • 60H15
  • 60F17
  • 60J60
  • 60G09
  • 60K40
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