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Improvement of Wu's logarithmic Sobolev inequality via the Poisson-Föllmer process

Shrey Aryan, Pablo López-Rivera, Yair Shenfeld

TL;DR

We address Wu's logarithmic Sobolev inequality for the Poisson measure on the nonnegative integers by deriving a stochastic entropy representation via the Poisson-Föllmer process, mirroring the Gaussian approach of Lehec. This yields a concise, probabilistic proof of Wu's inequality and a deficit identity that enables a quantitative improvement under ultra-log-concavity, with an explicit lower bound δ(f) ≥ (T^2/2) Θ_{f(0)/f(1)}(E[μ]/T). The improvement is expressed through the function Θ_c(z) = z^2/(1+cz) log(1/(1+cz)) − z^2/(1+cz) + z^2, which is nonnegative and connected to relative entropy between Poisson measures of differing intensities. Equality in Wu's inequality is shown to occur only for exponential profiles, and the work situates the discrete result within the broader Gaussian setting, highlighting both parallels and discrete-specific challenges.

Abstract

We give an alternative proof to Wu's logarithmic Sobolev inequality for the Poisson measure on the nonnegative integers using a stochastic variational formula for entropy. We show that this approach leads to improvement of Wu's inequality under convexity assumptions.

Improvement of Wu's logarithmic Sobolev inequality via the Poisson-Föllmer process

TL;DR

We address Wu's logarithmic Sobolev inequality for the Poisson measure on the nonnegative integers by deriving a stochastic entropy representation via the Poisson-Föllmer process, mirroring the Gaussian approach of Lehec. This yields a concise, probabilistic proof of Wu's inequality and a deficit identity that enables a quantitative improvement under ultra-log-concavity, with an explicit lower bound δ(f) ≥ (T^2/2) Θ_{f(0)/f(1)}(E[μ]/T). The improvement is expressed through the function Θ_c(z) = z^2/(1+cz) log(1/(1+cz)) − z^2/(1+cz) + z^2, which is nonnegative and connected to relative entropy between Poisson measures of differing intensities. Equality in Wu's inequality is shown to occur only for exponential profiles, and the work situates the discrete result within the broader Gaussian setting, highlighting both parallels and discrete-specific challenges.

Abstract

We give an alternative proof to Wu's logarithmic Sobolev inequality for the Poisson measure on the nonnegative integers using a stochastic variational formula for entropy. We show that this approach leads to improvement of Wu's inequality under convexity assumptions.
Paper Structure (11 sections, 15 theorems, 83 equations, 1 figure, 1 table)

This paper contains 11 sections, 15 theorems, 83 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. The Poisson-Föllmer process
  4. Preliminaries
  5. The Poisson-Föllmer process
  6. Improvement of Wu's inequality
  7. Proof of Wu's inequality via the Poisson-Föllmer process
  8. An identity for the deficit in Wu's inequality
  9. Improvement of Wu's inequality under ultra-log-concavity
  10. The equality cases of Wu's inequality
  11. Comparison with the Gaussian setting

Key Result

Theorem 1.1

Fix $T>0$, and let $\pi_T$ be the Poisson measure on $\mathbb N$ with intensity $T$. Let $f:\mathbb N\to (0,\infty)$ be an $L^1(\pi_T)$-integrable function, and define Then,

Figures (1)

  • Figure 2.1: The points in $[0,T]\times \mathbb R_{\ge 0}$ 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 (26)

  • Theorem 1.1: Wu's inequality MR1800540
  • Theorem 1.2: Improvement of Wu's inequality under ultra-log-concavity
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 16 more