[Convergence en distance de Wasserstein pour une diffusion auto-stabilisante évoluant dans un paysage à double puits]
Il est bien connu (voir Bolley et al. [3]) qu'il existe une contraction en distance de Wasserstein entre la solution de l'équation des milieux granulaires et son unique état d'équilibre, pour peu que le potentiel de confinement soit strictement convexe. Néanmoins, dans le cas non convexe, on dispose de peu de résultats. En particulier, sous des conditions simples à vérifier, il n'y a pas unicité de l'état d'équilibre. Par conséquent, la méthode de Bolley, Gentil et Guillin ne peut pas être appliquée sous ces conditions. Toutefois, ici, nous présentons un exemple simple (par souci de simplicité) d'un potentiel de confinement à deux puits, et nous montrons la convergence vers 0 de la distance de Wasserstein entre la solution de l'équation des milieux granulaires et une application (qui caractérise les états d'équilibre) de cette solution.
It is well-known (see Bolley et al. [3]) that there exists a contraction in Wasserstein distance between the solution to the granular media equation and its unique steady state, provided that the confining potential is strictly convex. Nevertheless, in the nonconvex case, just few is known. In particular, we do not have a unique steady state under easily checked assumptions if the diffusion coefficient is sufficiently small. Consequently, the method of Bolley, Gentil and Guillin can not be applied in this setting. However, here, we present a simple example (for the sake of the simplicity) of a double-well confining potential, and we show the convergence to 0 of the Wasserstein distance between the solution to the granular media equation and a related application (which characterizes the steady states) of this solution.
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@article{CRMATH_2018__356_6_657_0,
author = {Tugaut, Julian},
title = {Convergence in {Wasserstein} distance for self-stabilizing diffusion evolving in a double-well landscape},
journal = {Comptes Rendus. Math\'ematique},
pages = {657--660},
publisher = {Elsevier},
volume = {356},
number = {6},
year = {2018},
doi = {10.1016/j.crma.2018.04.020},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2018.04.020/}
}
TY - JOUR AU - Tugaut, Julian TI - Convergence in Wasserstein distance for self-stabilizing diffusion evolving in a double-well landscape JO - Comptes Rendus. Mathématique PY - 2018 SP - 657 EP - 660 VL - 356 IS - 6 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2018.04.020/ DO - 10.1016/j.crma.2018.04.020 LA - en ID - CRMATH_2018__356_6_657_0 ER -
%0 Journal Article %A Tugaut, Julian %T Convergence in Wasserstein distance for self-stabilizing diffusion evolving in a double-well landscape %J Comptes Rendus. Mathématique %D 2018 %P 657-660 %V 356 %N 6 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2018.04.020/ %R 10.1016/j.crma.2018.04.020 %G en %F CRMATH_2018__356_6_657_0
Tugaut, Julian. Convergence in Wasserstein distance for self-stabilizing diffusion evolving in a double-well landscape. Comptes Rendus. Mathématique, Tome 356 (2018) no. 6, pp. 657-660. doi : 10.1016/j.crma.2018.04.020. http://archive.numdam.org/articles/10.1016/j.crma.2018.04.020/
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