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Probabilistic approach for granular media equations in the non-uniformly convex case
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  • Published: 15 March 2007

Probabilistic approach for granular media equations in the non-uniformly convex case

  • P. Cattiaux1,2,
  • A. Guillin3 &
  • F. Malrieu4 

Probability Theory and Related Fields volume 140, pages 19–40 (2008)Cite this article

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Abstract

We use here a particle system to prove both a convergence result (with convergence rate) and a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. The proof of convergence is simpler than the one in Carrillo–McCann–Villani (Rev. Mat. Iberoamericana 19:971–1018, 2003; Arch. Rat. Mech. Anal. 179:217–263, 2006). All the results complete former results of Malrieu (Ann. Appl. Probab. 13:540–560, 2003) in the uniformly convex case. The main tool is an uniform propagation of chaos property and a direct control in Wasserstein distance of solutions starting with different initial measures. The deviation inequality is obtained via a T 1 transportation cost inequality replacing the logarithmic Sobolev inequality which is no more clearly dimension free.

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Authors and Affiliations

  1. École Polytechnique, CMAP, Route de Saclay, 91128, Palaiseau Cedex, France

    P. Cattiaux

  2. Université Paris X Nanterre, Equipe MODAL’X, UFR SEGMI, 200 avenue de la République, 92001, Nanterre Cedex, France

    P. Cattiaux

  3. CEREMADE, UMR CNRS 7534, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France

    A. Guillin

  4. IRMAR, Université Rennes 1, Campus de Beaulieu, 35042, Rennes Cedex, France

    F. Malrieu

Authors
  1. P. Cattiaux
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  2. A. Guillin
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  3. F. Malrieu
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Correspondence to P. Cattiaux.

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Cattiaux, P., Guillin, A. & Malrieu, F. Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Relat. Fields 140, 19–40 (2008). https://doi.org/10.1007/s00440-007-0056-3

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  • Received: 29 October 2006

  • Published: 15 March 2007

  • Issue Date: January 2008

  • DOI: https://doi.org/10.1007/s00440-007-0056-3

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Keywords

  • Granular media equation
  • Transportation cost inequality
  • Logarithmic Sobolev Inequalities
  • Concentration inequalities

Mathematics Subject Classification (2000)

  • 65C35
  • 35K55
  • 65C05
  • 82C22
  • 26D10
  • 60E15
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