Wasserstein Stability of the Entropy Power Inequality for Log-Concave Densities

Abstract

We establish quantitative stability results for the entropy power inequality (EPI). Specifically, we show that if uniformly log-concave densities nearly saturate the EPI, then they must be close to Gaussian densities in the quadratic Wasserstein distance. Further, if one of the densities is log-concave and the other is Gaussian, then the deficit in the EPI can be controlled in terms of the $L^1$-Wasserstein distance. As a counterpoint, an example shows that the EPI can be unstable with respect to the quadratic Wasserstein distance when densities are uniformly log-concave on sets of measure arbitrarily close to one. Our stability results can be extended to non-log-concave densities, provided certain regularity conditions are met. The proofs are based on optimal transportation.