Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker-Planck equations

TL;DR

This paper introduces discrete models of one-dimensional Fokker-Planck equations, proving exponential convergence to equilibrium using novel discrete inequalities and hypocoercive techniques, applicable to both continuous and discrete time settings.

Contribution

It develops new discrete Poincaré inequalities and adapts hypocoercive methods for discrete Fokker-Planck equations, advancing the understanding of their convergence properties.

Findings

Proves exponential decay to equilibrium for discrete Fokker-Planck equations.

Introduces new discrete Poincaré inequalities for velocity discretization.

Extends hypocoercive methods to discrete inhomogeneous problems.

Abstract

In this article, we propose and study several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker-Planck equations. In particular, for these discretizations of velocity and space, we prove the exponential convergence to the equilibrium of the solutions, for time-continuous equations as well as for time-discrete equations. Our method uses new types of discrete Poincar\'e inequalities for a "two-direction" discretization of the derivative in velocity. For the inhomogeneous problem, we adapt hypocoercive methods to the discrete cases.