A martingale bound for the entropy associated with a trimmed filtration on $\mathbb {R}^d$

TL;DR

This paper develops martingale-based bounds for the entropy of probability measures on Euclidean spaces, leading to a weighted log-Sobolev inequality in one dimension, advancing understanding of entropy inequalities.

Contribution

It introduces a novel martingale approach to bound entropy on  spaces, providing new integral bounds and a weighted log-Sobolev inequality in the one-dimensional case.

Findings

Derived entropy bounds using martingale methods

Established a weighted log-Sobolev inequality in 1D

Provided integral form bounds for probability measures

Abstract

Using martingale methods, we provide bounds for the entropy of a probability measure on $\mathbb {R}^d$ with the right-hand side given in a certain integral form. As a corollary, in the one-dimensional case, we obtain a weighted log-Sobolev inequality.