Fractional Fokker–Planck Equation with General Confinement Force

Équation de Fokker–Planck fractionnaire avec une force confinante générale

Laurent Lafleche (1, 2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 CMLS - Centre de Mathématiques Laurent Schwartz

This article studies a Fokker-Planck type equation of fractional diffusion with conservative drift ∂f/∂t = ∆^(α/2) f + div(Ef), where ∆^(α/2) denotes the fractional Laplacian and E is a confining force field. The main interest of the present paper is that it applies to a wide variety of force fields, with a few local regularity and a polynomial growth at infinity. We first prove the existence and uniqueness of a solution in weighted Lebesgue spaces depending on E under the form of a strongly continuous semigroup. We also prove the existence and uniqueness of a stationary state, by using an appropriate splitting of the fractional Laplacian and by proving a weak and strong maximum principle. We then study the rate of convergence to equilibrium of the solution. The semigroup has a property of regularization in fractional Sobolev spaces, as well as a gain of integrability and positivity which we use to obtain polynomial or exponential convergence to equilibrium in weighted Lebesgue spaces.

Mots clés: Fokker-Planck; fractional diffusion with drift; fractional Laplacian; confinement force; asymptotic behavior

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]; Analyse fonctionnelle [math.FA]

Laurent Lafleche. Fractional Fokker–Planck Equation with General Confinement Force. SIAM Journal on Mathematical Analysis, 2020, 52 (1), pp.164-196. ⟨10.1137/18M1188331⟩. ⟨hal-01724754⟩