Fractional Fokker-Planck Equation with General Confinement Force
Lafleche, Laurent (2020), Fractional Fokker-Planck Equation with General Confinement Force, SIAM Journal on Mathematical Analysis, 52, 1, p. 31. 10.1137/18M1188331
TypeArticle accepté pour publication ou publié
Journal nameSIAM Journal on Mathematical Analysis
SIAM - Society for Industrial and Applied Mathematics
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)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.
Subjects / Keywordsfractional Laplacian; Fokker-Planck; fractional diffusion with drift; confinement force; asymptotic behavior
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