Intertwining relations for one-dimensional diffusions and application to functional inequalities

TL;DR

This paper develops new intertwining relations for one-dimensional diffusion semigroups, enabling novel proofs of spectral gap formulas and criteria for logarithmic Sobolev inequalities, with improved estimates on constants for classical examples.

Contribution

It extends recent discrete case results to continuous diffusions, providing new tools for functional inequalities and spectral analysis.

Findings

New intertwining relations for one-dimensional diffusions

A new criterion for the logarithmic Sobolev inequality

Improved estimates on optimal constants in classical examples

Abstract

Following the recent work [13] fulfilled in the discrete case, we pro- vide in this paper new intertwining relations for semigroups of one-dimensional diffusions. Various applications of these results are investigated, among them the famous variational formula of the spectral gap derived by Chen and Wang [15] together with a new criterion ensuring that the logarithmic Sobolev inequality holds. We complete this work by revisiting some classical examples, for which new estimates on the optimal constants are derived.