Fractional Fokker-Planck Equation with General Confinement Force

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],

References in corpus (7)

• Fractional diffusion limit for collisional kinetic equations
• Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials
• Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres
• Regularity for parabolic integro-differential equations with very irregular kernels
• On the loss of continuity for super-critical drift-diffusion equations
• Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media
• Spectral analysis of semigroups and growth-fragmentation equations

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