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We prove stability estimates for the Shannon-Stam inequality (also known asthe entropy-power inequality) for log-concave random vectors in terms ofentropy and transportation distance. In particular, we give the first stabilityestimate for general log-concave random vectors in the following form: forlog-concave random vectors $X,Y \in \mathbb{R}^d$, the deficit in theShannon-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 withrespect to the standard Gaussian and the constant $C$ depends only on thecovariance structures and the spectral gaps of $X$ and $Y$. In the case ofuniformly log-concave vectors our analysis gives dimension-free bounds. Ourproofs are based on a new approach which uses an entropy-minimizing processfrom stochastic control theory.

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