Probability Theory and Related Fields

, Volume 140, Issue 1–2, pp 19–40 | Cite as

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



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.


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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© Springer-Verlag 2007

Authors and Affiliations

  1. 1.École Polytechnique, CMAPPalaiseau CedexFrance
  2. 2.Université Paris X Nanterre, Equipe MODAL’X, UFR SEGMINanterre CedexFrance
  3. 3.CEREMADE, UMR CNRS 7534Paris Cedex 16France
  4. 4.IRMAR, Université Rennes 1Rennes CedexFrance

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