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

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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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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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Keywords

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

Mathematics Subject Classification (2000)

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