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

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-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 logconcave 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.