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

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We use here a particle system to prove a convergence result as well as a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. Proof is straightforward, simplifying deeply proofs of Carrillo-McCann-Villani \cite{CMV,CMV2} and completing results of Malrieu \cite{malrieu03} in the uniformly convex case. It relies on 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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Probability, 65C35, 35K55, 65C05, 82C22, 26D10, 60E15

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http://salesiana.dossiersoluciones.com/handle/123456789/121535

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0603541
 http://arxiv.org/abs/math/0603541

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arXiv