Fractional Fokker-Planck Equation with General Confinement Force

TL;DR

This paper investigates a fractional Fokker-Planck equation with a general confining force, establishing existence, uniqueness, and convergence rates of solutions, and analyzing their regularization and positivity properties in weighted Lebesgue and Sobolev spaces.

Contribution

It extends the analysis of fractional Fokker-Planck equations to a broad class of confining forces with minimal regularity and growth conditions, proving well-posedness and convergence to equilibrium.

Findings

Existence and uniqueness of solutions in weighted Lebesgue spaces.

Existence and uniqueness of stationary states.

Polynomial and exponential convergence to equilibrium.

Abstract

This article studies a Fokker-Planck type equation of fractional diffusion with conservative drift $\partial$f/$\partial$t = $\Delta$^($\alpha$/2) f + div(Ef), where $\Delta$^($\alpha$/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. Mathematics Subject Classification (2000): 47D06 One-parameter semi-groups and linear evolution equations [See also 34G10, 34K30], 35P15 Estimation of eigenvalues, upper and lower bounds [See also 35P05, 45C05, 47A10], 35B40 Partial differential equations, Asymptotic behavior of solutions [see also 45C05, 45K05, 35410],