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

被引：98
作者：
Cattiaux, P.
Guillin, A.
Malrieu, F.
机构：
[1] Ecole Polytech, CMAP, F-91128 Palaiseau, France
[2] Univ Paris 10, Equipe MODALX, UFR, SEGMI, F-92001 Nanterre, France
[3] Univ Paris 09, CEREMADE, CNRS, UMR 7534, F-75775 Paris, France
[4] Univ Rennes 1, IRMAR, F-35042 Rennes, France
来源：
PROBABILITY THEORY AND RELATED FIELDS | 2008年 / 140卷 / 1-2期
关键词：
granular media equation; transportation cost inequality; logarithmic Sobolev inequalities; concentration inequalities;
D O I：
10.1007/s00440-007-0056-3
中图分类号：
O21 [概率论与数理统计]; C8 [统计学];
学科分类号：
020208 ; 070103 ; 0714 ;
摘要：
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.
引用
收藏
页码：19 / 40
页数：22
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