Quantitative logarithmic Sobolev inequalities and stability estimates. (English) Zbl 1355.60094
Summary: We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincaré inequality. The result implies a lower bound on the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigate the deficit in the Talagrand quadratic transportation cost inequality this time by means of an \(L^1\)-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 context of the Bakry-Émery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.
MSC:
| 60H99 | Stochastic analysis |
| 60J60 | Diffusion processes |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
logarithmic Sobolev inequalities; Gaussian measure; probability densities; Poincaré inequality; deficit estimates; transportation inequalities; optimal transport; semigroup theoryReferences:
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