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November 2018 Convergence to equilibrium in the free Fokker–Planck equation with a double-well potential
Catherine Donati-Martin, Benjamin Groux, Mylène Maïda
Ann. Inst. H. Poincaré Probab. Statist. 54(4): 1805-1818 (November 2018). DOI: 10.1214/17-AIHP856

Abstract

We consider the one-dimensional free Fokker–Planck equation

\[\frac{\partial \mu_{t}}{\partial t}=\frac{\partial }{\partial x}[\mu_{t}\cdot (\frac{1}{2}V'-H\mu_{t})],\] where $H$ denotes the Hilbert transform and $V$ is a particular double-well quartic potential, namely $V(x)=\frac{1}{4}x^{4}+\frac{c}{2}x^{2}$, with $c\ge -2$. We prove that the solution $(\mu_{t})_{t\ge 0}$ of this PDE converges in Wasserstein distance of any order $p\ge 1$ to the equilibrium measure $\mu_{V}$ as $t$ goes to infinity. This provides a first result of convergence for this equation in a non-convex setting. The proof involves free probability and complex analysis techniques.

On considère l’équation de Fokker–Planck libre unidimensionnelle

\[\frac{\partial \mu_{t}}{\partial t}=\frac{\partial }{\partial x}[\mu_{t}\cdot (\frac{1}{2}V'-H\mu_{t})],\] où $H$ désigne la transformée de Hilbert et $V$ est un potentiel quartique à double puits particulier, à savoir $V(x)=\frac{1}{4}x^{4}+\frac{c}{2}x^{2}$ avec $c\ge -2$. On démontre que la solution $(\mu_{t})_{t\ge 0}$ de cette EDP converge pour une distance de Wasserstein d’ordre quelconque $p\ge 1$ vers la mesure d’équilibre $\mu_{V}$ quand $t$ tend vers l’infini. Cela fournit un premier résultat de convergence pour cette équation dans un cadre non convexe. La démonstration fait intervenir les probabilités libres et l’analyse complexe.

Citation

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Catherine Donati-Martin. Benjamin Groux. Mylène Maïda. "Convergence to equilibrium in the free Fokker–Planck equation with a double-well potential." Ann. Inst. H. Poincaré Probab. Statist. 54 (4) 1805 - 1818, November 2018. https://doi.org/10.1214/17-AIHP856

Information

Received: 11 August 2016; Revised: 18 July 2017; Accepted: 27 July 2017; Published: November 2018
First available in Project Euclid: 18 October 2018

zbMATH: 06996550
MathSciNet: MR3865658
Digital Object Identifier: 10.1214/17-AIHP856

Subjects:
Primary: 35B40, 46L54, 60B20

Rights: Copyright © 2018 Institut Henri Poincaré

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Vol.54 • No. 4 • November 2018
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