Convergence to equilibrium for Fokker-Planck equation with a general force field

TL;DR

This paper investigates the long-term convergence of solutions to the Fokker-Planck equation with a general force field, not necessarily derived from a potential, using operator splitting and Krein-Rutmann theory.

Contribution

It extends convergence results to Fokker-Planck equations with non-potential force fields through a novel operator splitting approach.

Findings

Proves convergence to equilibrium for general force fields.

Utilizes Krein-Rutmann theory in the analysis.

Provides a framework for non-potential force field cases.

Abstract

The aim of this paper is to study the convergence of the solution of the Fokker-Planck equation to the associated stationary state when time goes to infinity. The force field which we consider here is of a general structure, that is it may not derive from a potential. The proof is based on an adequate splitting L = B+A of the Fokker-Planck operator L and the use of Krein-Rutmann theory.