The Poisson transport map

TL;DR

The paper constructs a Poisson transport map from the Poisson point process to ultra-log-concave measures on $\mathbb N$ and proves it is a $1$-Lipschitz contraction in the discrete setting. This enables transferring functional inequalities from the Poisson space to UL-concave measures, overcoming discrete-chain-rule obstacles and yielding sharp constants in modified logarithmic Sobolev inequalities, Phi-Sobolev inequalities, and transport-entropy inequalities. The approach parallels the Brownian transport framework but adapts to discrete structure via a Poisson-based Föllmer-type process and a detailed Malliavin calculus on the Poisson space. Consequently, the authors obtain improved constants and a coherent suite of concentration and entropy estimates for UL-concave measures, broadening the applicability of transport-inequality methods to combinatorial and discrete-probability contexts.

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.

Figures (1)

• Figure 1.1: The points in $[0,T]\times [0,M]$ are generated according to a standard Poisson process (7 points in this case). At time $t\in [0,T]$ the value of the process $X_t^{\lambda}$ is equal to the number of points under the curve (filled circles). In the figure, $X_2^{\lambda}=1$ and $X_{T}^{\lambda}=4$.

Theorems & Definitions (40)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6: The optimal constant
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Proposition 2.2