The Fokker-Planck equation with subcritical confinement force

TL;DR

This paper studies the Fokker-Planck equation with a broad class of confinement forces, proving existence of equilibrium and establishing improved convergence rates under more general conditions.

Contribution

It extends previous results by allowing more general force fields and spaces, and provides sharper convergence rates to equilibrium.

Findings

Existence of equilibrium for general force fields

Polynomial and stretch exponential convergence rates

Improved results over prior work by Toscani, Villani, Röchner, Wang

Abstract

We consider the Fokker-Planck equation with subcritical confinement force field which may not derive from a potential function. We prove the existence of an equilibrium (in the case of a general force) and we establish some (polynomial and stretch exponential) rate of convergence to the equilibrium (depending on the space to which belongs the initial datum). Our results improve similar results established by Toscani, Villani [29] and R{\"o}chner , Wang [27]: the force field is more general, the spaces are more general, the rates are sharper.