Comptes Rendus
no. 6
Partial differential equations/Probability theory
Convergence in Wasserstein distance for self-stabilizing diffusion evolving in a double-well landscape

Presented by: Haïm Brézis

Julian Tugaut1
1 Univ Lyon, Université Jean-Monnet, CNRS UMR 5208, Institut Camille-Jordan, Maison de l'Université, 10, rue Tréfilerie, CS 82301, 42023 Saint-Étienne cedex 2, France
Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 657-660.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.04.020

Julian Tugaut 1

1 Univ Lyon, Université Jean-Monnet, CNRS UMR 5208, Institut Camille-Jordan, Maison de l'Université, 10, rue Tréfilerie, CS 82301, 42023 Saint-Étienne cedex 2, France 
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Julian Tugaut. Convergence in Wasserstein distance for self-stabilizing diffusion evolving in a double-well landscape. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 657-660. doi : 10.1016/j.crma.2018.04.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.020/

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