2006/03/22 by Patrick Cattiaux, Arnaud Guillin, Cattiaux, Patrick +3 · 6 citations
Environmental Science · Mathematics · Engineering · #Landslides and related hazards #Numerical methods in inverse problems #Soil and Unsaturated Flow
paper · doi:10.48550/arxiv.math/0603541
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 \citeCMV,CMV2 and completing results of Malrieu \citemalrieu03 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.