On the discrete Poincar\'e inequality for B-schemes of 1D Fokker-Planck equations in full space

TL;DR

This paper develops two methods to establish the discrete Poincaré inequality for B-schemes discretizations of the 1D Fokker-Planck equation, ensuring exponential convergence to equilibrium.

Contribution

It introduces Gamma-calculus and Lyapunov function approaches to derive the discrete Poincaré inequality for B-schemes, extending the analysis to full space.

Findings

Discrete Poincaré inequality established for B-schemes

Exponential convergence to equilibrium demonstrated

Methods potentially extendable to higher dimensions

Abstract

In this paper, we propose two approaches to derive the discrete Poincar\'e inequality for the B-schemes, a family of finite volume discretization schemes, for the one-dimensional Fokker-Planck equation in full space. We study the properties of the spatially discretized Fokker-Planck equation in the viewpoint of a continuous-time Markov chain. The first approach is based on Gamma-calculus, through which we show that the Bakry-\'Emery criterion still holds in the discrete setting. The second approach employs the Lyapunov function method, allowing us to extend a local discrete Poincar\'e inequality to the full space. The assumptions required for both approaches are roughly comparable with some minor differences. These methods have the potential to be extended to higher dimensions. As a result, we obtain exponential convergence to equilibrium for the discrete schemes by applying the discrete Poincar\'e inequality.