Fokker-Planck equations and one-dimensional functional inequalities for heavy tailed densities

TL;DR

This paper derives sharp functional inequalities like Poincaré and logarithmic Sobolev for heavy-tailed distributions such as inverse Gamma and Cauchy, using Fokker-Planck equations with these densities as steady states.

Contribution

It introduces improved weighted inequalities for heavy-tailed densities, extending prior results with better weight functions for Cauchy-type distributions.

Findings

Established sharp inequalities for inverse Gamma densities.

Improved logarithmic Sobolev inequalities for Cauchy-type densities.

Utilized Fokker-Planck equations to derive these inequalities.

Abstract

We study one-dimensional functional inequalities of the type of Poincar\'e, logarithmic Sobolev and Wirtinger, with weight, for probability densities with polynomial tails. As main examples, we obtain sharp inequalities satisfied by inverse Gamma densities, taking values on $R_+$, and Cauchy-type densities, taking values on $R$. In this last case, we improve the result obtained by Bobkov and Ledoux in 2009 by introducing a better weight function in the logarithmic Sobolev inequality. The results are obtained by resorting to Fokker-Planck type equations which possess these densities as steady states.