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Existence and stability for Fokker–Planck equations with log-concave reference measure
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  • Published: 03 September 2008

Existence and stability for Fokker–Planck equations with log-concave reference measure

  • Luigi Ambrosio1,
  • Giuseppe Savaré2 &
  • Lorenzo Zambotti3 

Probability Theory and Related Fields volume 145, pages 517–564 (2009)Cite this article

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  • 58 Citations

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Abstract

We study Markov processes associated with stochastic differential equations, whose non-linearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability property: if the associated invariant measures converge weakly, then the Markov processes converge in law. The proofs are based on the interpretation of a Fokker–Planck equation as the steepest descent flow of the relative entropy functional in the space of probability measures, endowed with the Wasserstein distance.

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Author information

Authors and Affiliations

  1. Scuola Normale Superiore, Pisa, Italy

    Luigi Ambrosio

  2. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Giuseppe Savaré

  3. Université Pierre et Marie Curie, LPMA, 4 place Jussieu, 75252, Paris cedex 05, France

    Lorenzo Zambotti

Authors
  1. Luigi Ambrosio
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  2. Giuseppe Savaré
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  3. Lorenzo Zambotti
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Corresponding author

Correspondence to Lorenzo Zambotti.

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Ambrosio, L., Savaré, G. & Zambotti, L. Existence and stability for Fokker–Planck equations with log-concave reference measure. Probab. Theory Relat. Fields 145, 517–564 (2009). https://doi.org/10.1007/s00440-008-0177-3

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  • Received: 27 April 2007

  • Revised: 17 July 2008

  • Published: 03 September 2008

  • Issue Date: November 2009

  • DOI: https://doi.org/10.1007/s00440-008-0177-3

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Keywords

  • Reversible Markov processes
  • Log-concave probability measures
  • Gradient flows
  • Optimal transportation
  • Relative entropy

Mathematics Subject Classification (2000)

  • 60J60
  • 37L40
  • 60G07
  • 49Q20
  • 35K90
  • 28A33
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