Entropy-type inequalities for generalized Gamma densities

TL;DR

This paper studies how solutions to certain Fokker-Planck equations relax to equilibrium, characterized by generalized Gamma densities, and establishes new inequalities related to these densities.

Contribution

It introduces new weighted Poincaré and logarithmic Sobolev inequalities for generalized Gamma densities, linked to the relaxation analysis of Fokker-Planck equations.

Findings

Proved relaxation to equilibrium for the considered Fokker-Planck equations.

Derived new weighted inequalities for generalized Gamma densities.

Connected inequalities to the system's equilibrium behavior.

Abstract

We investigate the relaxation to equilibrium of the solution of a class of one-dimensional linear Fokker--Planck type equations that have been recently considered in connection with the study of addiction phenomena in a system of individuals. The steady states of these equations belong to the class of generalized Gamma densities. As a by-product of the relaxation analysis, we prove new weighted Poincar\'e and logarithmic Sobolev type inequalities for this class of densities.