Stability of the Shannon-Stam inequality via the Föllmer process

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.