Uniform semigroup spectral analysis of the discrete, fractional \& classical Fokker-Planck equations

TL;DR

This paper conducts a unified spectral analysis of discrete, fractional, and classical Fokker-Planck equations using semigroup theory, improving perturbation methods and establishing convergence and spectral estimates across different equation classes.

Contribution

It introduces a unified framework for spectral analysis of various Fokker-Planck equations, extending perturbation techniques and relaxing previous assumptions.

Findings

Established uniform spectral estimates for classical and discrete Fokker-Planck equations.

Extended spectral analysis to fractional Fokker-Planck equations and their discrete counterparts.

Improved perturbative methods to encompass a broader class of operators.

Abstract

In this paper, we investigate the spectral analysis (from the point of view of semi-groups) of discrete, fractional and classical Fokker-Planck equations. Discrete and fractional Fokker-Planck equations converge in some sense to the classical one. As a consequence, we first deal with discrete and classical Fokker-Planck equations in a same framework, proving uniform spectral estimates using a perturbation argument and an enlargement argument. Then, we do a similar analysis for fractional and classical Fokker-Planck equations using an argument of enlargement of the space in which the semigroup decays. We also handle another class of discrete Fokker-Planck equations which converge to the fractional Fokker-Planck one, we are also able to treat these equations in a same framework from the spectral analysis viewpoint, still with a semigroup approach and thanks to a perturbative argument combined with an enlargement one. Let us emphasize here that we improve the perturbative argument introduced in [7] and developed in [11], relaxing the hypothesis of the theorem, enlarging thus the class of operators which fulfills the assumptions required to apply it.