The Lévy–Fokker–Planck equation: Φ-entropies and convergence to equilibrium

Article type: Research Article

Authors: Gentil, Ivan | Imbert, Cyril

Affiliations: Centre de Recherche en Mathématiques de la Décision (CEREMADE, UMR CNRS 7534), Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris cedex 16, France. E-mails: {gentil, imbert}@ceremade.dauphine.fr

Abstract: In this paper, we study a Fokker–Planck equation of the form ut=ℐ[u]+div(xu), where the operator ℐ, which is usually the Laplacian, is replaced here with a general Lévy operator. We prove by the entropy production method the exponential decay in time of the solution to the steady state of the associated stationary equation.

Keywords: Fokker–Planck equation, Lévy operator, Φ-entropy inequalities, entropy production method, logarithmic Sobolev inequalities, fractional Laplacian

DOI: 10.3233/ASY-2008-0887

Journal: Asymptotic Analysis, vol. 59, no. 3-4, pp. 125-138, 2008