The Kinetic Fokker-Planck Equation with General Force

TL;DR

This paper studies the kinetic Fokker-Planck equation with general forces, proving existence, uniqueness, and explicit exponential convergence rates to equilibrium, extending classical results to more general forces.

Contribution

It extends classical force results to general forces and provides explicit convergence rates, improving understanding of the kinetic Fokker-Planck equation.

Findings

Existence and uniqueness of equilibrium for general forces

Explicit exponential convergence rates to equilibrium

Improved convergence rate for Fitzhugh-Nagumo equation

Abstract

We consider the kinetic Fokker-Planck equation with a class of general force. We prove the existence and uniqueness of a positive normalized equilibrium (in the case of a general force) and establish some exponential rate of convergence to the equilibrium (and the rate can be explicitly computed). Our results improve results about classical force to general force case. Our result also improve the rate of convergence for the Fitzhugh-Nagumo equation from non-quantitative to quantitative explicit rate.