The Poisson transport map

TL;DR

This paper constructs a contraction transport map from Poisson processes to ultra-log-concave measures on natural numbers, enabling new functional inequalities and improving constants in modified logarithmic Sobolev inequalities.

Contribution

It introduces a novel transport map in discrete settings that overcomes previous obstacles, leading to improved inequalities for ultra-log-concave measures.

Findings

Established a contraction transport map from Poisson processes to ultra-log-concave measures

Derived the best known constants in modified logarithmic Sobolev inequalities for these measures

Enabled transfer of functional inequalities in discrete probability spaces

Abstract

We construct a transport map from Poisson point processes onto ultra-log-concave measures over the natural numbers, and show that this map is a contraction. Our approach overcomes the known obstacles to transferring functional inequalities using transport maps in discrete settings, and allows us to deduce a number of functional inequalities for ultra-log-concave measures. In particular, we provide the currently best known constant in modified logarithmic Sobolev inequalities for ultra-log-concave measures.