Logarithmic Sobolev inequalities: regularizing effect of L\'evy operators and asymptotic convergence in the L\'evy-Fokker-Planck equation

TL;DR

This paper explores how Lévý logarithmic Sobolev inequalities influence the regularity of solutions to fractional heat equations and the long-term behavior of Lévý-Ornstein-Uhlenbeck processes.

Contribution

It applies Lévý logarithmic Sobolev inequalities to analyze regularity and asymptotic convergence in fractional heat equations and Lévý stochastic processes.

Findings

Enhanced understanding of solution regularity for fractional heat equations

Insights into the asymptotic behavior of Lévý-Ornstein-Uhlenbeck processes

Application of inequalities to stochastic process convergence

Abstract

In this paper we study some applications of the L\'evy logarithmic Sobolev inequality to the study of the regularity of the solution of the fractal heat equation, i. e. the heat equation where the Laplacian is replaced with the fractional Laplacian. It is also used to the study of the asymptotic behaviour of the L\'evy-Ornstein-Uhlenbeck process.