# Stability of the Shannon-Stam inequality via the F\"ollmer process

## TL;DR

This paper establishes stability estimates for the Shannon-Stam inequality for log-concave vectors, linking the deficit to entropy and transportation metrics, using a novel stochastic control approach.

## Contribution

It provides the first stability estimates for general log-concave vectors in the Shannon-Stam inequality, with dimension-free bounds for uniformly log-concave cases.

## Key findings

- Bound the Shannon-Stam deficit by relative entropy with respect to Gaussian.
- First stability estimate for general log-concave vectors.
- Dimension-free bounds for uniformly log-concave vectors.

## Abstract

We prove stability estimates for the Shannon-Stam inequality (also known as the entropy-power inequality) for log-concave random vectors in terms of entropy and transportation distance. In particular, we give the first stability estimate for general log-concave random vectors in the following form: for log-concave random vectors $X,Y \in \mathbb{R}^d$, the deficit in the Shannon-Stam inequality is bounded from below by the expression $$ C \left(\mathrm{D}\left(X||G\right) + \mathrm{D}\left(Y||G\right)\right), $$ where $\mathrm{D}\left( \cdot ~ ||G\right)$ denotes the relative entropy with respect to the standard Gaussian and the constant $C$ depends only on the covariance structures and the spectral gaps of $X$ and $Y$. In the case of uniformly log-concave vectors our analysis gives dimension-free bounds. Our proofs are based on a new approach which uses an entropy-minimizing process from stochastic control theory.