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

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.

Title changed. Added Section 4 on equality cases. Made some minor corrections in Remark 1.4 and proof of Lemma 3.6