Fractional Fokker-Planck equation

TL;DR

This paper studies the long-term behavior of solutions to a fractional Fokker-Planck equation, proving exponential convergence to equilibrium in a weighted L^1 space using advanced functional analysis techniques.

Contribution

It improves previous results by establishing convergence in a weighted L^1 space, expanding the understanding of the equation's long-time behavior.

Findings

Proves exponential convergence towards equilibrium in weighted L^1 space.

Extends previous L^2 space results to more general weighted spaces.

Utilizes recent abstract theory to enhance functional space analysis.

Abstract

This paper deals with the long time behavior of solutions to a "fractional Fokker-Planck" equation of the form $\partial_t f = I[f] + \text{div}(xf)$ where the operator $I$ stands for a fractional Laplacian. We prove an exponential in time convergence towards equilibrium in new spaces. Indeed, such a result was already obtained in a $L^2$ space with a weight prescribed by the equilibrium in \cite{GI}. We improve this result obtaining the convergence in a $L^1$ space with a polynomial weight. To do that, we take advantage of the recent paper \cite{GMM} in which an abstract theory of enlargement of the functional space of the semigroup decay is developed.