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

Malrieu, Florent; Guillin, Arnaud; Cattiaux, Patrick (2008), Probabilistic approach for granular media equations in the non uniformly convex case, Probability Theory and Related Fields, 140, 1-2, p. 19-40. http://dx.doi.org/10.1007/s00440-007-0056-3

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00021591/en/
Date
2008
Journal name
Probability Theory and Related Fields
Volume
140
Number
1-2
Publisher
Springer
Pages
19-40
Publication identifier
http://dx.doi.org/10.1007/s00440-007-0056-3
Metadata
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Author(s)
Malrieu, Florent
Guillin, Arnaud
Cattiaux, Patrick
Abstract (EN)
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.
Subjects / Keywords
Concentration inequalities; Logarithmic Sobolev Inequalities; transportation cost inequality; Granular media equation

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