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

Cyril Imbert (1) , Ivan Gentil (2)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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

Mots clés: Fokker-Planck equation; Ornstein-Uhlenbeck equation; Lévy process; Phi-entropy inequalities; entropy production method; logarithmic Sobolev inequalities; fractional Laplacian

Domaines: Equations aux dérivées partielles [math.AP]

Cyril Imbert, Ivan Gentil. The Lévy-Fokker-Planck equation: Phi-entropies and convergence to equilibrium. 2006. ⟨hal-00113806v1⟩