Quantitative logarithmic Sobolev inequalities and stability estimates

We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussianmeasure restricted to probability densities which satisfy a Poincaré inequality.The result implies a lower boundon the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigatethe deficit in the Talagrand quadratic transportation cost inequality this time by means of anL¹-Kantorovich-Wasserstein distance, optimal for product measures, and deduce a lower bound on the deficit in the logarithmic Sobolev inequality in terms of this metric. Applications are given in the contextof the Bakry-Émery theory and the coherent state transform. The proofs combine tools fromsemigroup and heat kernel theory and optimal mass transportation.