Representation formula for the entropy and functional inequalities

TL;DR

This paper introduces a stochastic formula for Gaussian relative entropy, enabling simplified proofs of various functional inequalities, thus advancing theoretical understanding and proof techniques in the field.

Contribution

It provides a new stochastic representation for Gaussian relative entropy, facilitating unified proofs of multiple functional inequalities.

Findings

Derived a stochastic formula for Gaussian relative entropy.

Unified proofs for several functional inequalities.

Enhanced understanding of entropy and inequalities through probabilistic methods.

Abstract

We prove a stochastic formula for the Gaussian relative entropy in the spirit of Borell's formula for the Laplace transform. As an application, we give unified and short proofs of a number of functional inequalities.