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

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

2006-03-22

arxiv.org/abs/math/0603541v1

Mathematics

Probability

Scientific paper

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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