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

Stéphane Mischler,Isabelle Tristani

doi:10.5802/jep.46

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.

Highlights

In order to write a rough version of our main result, we introduce some notation
Λε = Aε + Bε, where Bε enjoys suitable dissipativity property and Aε enjoys some suitable Bε-power regularity (a property that we introduce in Section 2.4 and that we name in that way by analogy with the Bε-power compactness notion introduced by Voigt [16])
Uniform Bε-power regularity of Aε. — we prove that AεSBε and its iterated convolution products fulfill nice regularization and growth estimates

Summary

Introduction

— In this paper, we are interested in the spectral analysis and the long time asymptotic convergence of semigroups associated to some discrete, fractional and classical Fokker-Planck equations. The discrete models can be seen as (singular) perturbations of the limit equations and our analysis takes advantage of such a property in order to capture the asymptotic behaviour of the related spectral objects (spectrum, spectral projector) and to conclude the above uniform spectral decomposition. This kind of perturbative method has been introduced in [8] and improved in [15].

Comments and possible extensions

From discrete to classical Fokker-Planck equation

From fractional to classical Fokker-Planck equation

Regularization properties of Aε

Dissipativity properties of Bε and B0