Quantitative logarithmic Sobolev inequalities and stability estimates
By Max Fathi, E. Indrei, Michel Ledoux. 2014
88 citations recorded by the source corpus.
We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincar\'e 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 ${\rm L}^1$-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-\'Emery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.
Subfields: Applied Mathematics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/c77b9910828cd87cc996fa9b8e3426f7cd9b8f40
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In Applied Mathematics
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In Statistics and Probability
Citations in the corpus
- Stability Results for Logarithmic Sobolev and Gagliardo–Nirenberg Inequalities (2014, background): “(7) Inequality (4) follows by substracting 12 ∫ Rd |u|2 log |u|2 dμ from both sides of the inequality, which is more or less the idea that has been exploited by Fathi et al. [2014]. Consider a nonnegative function f ∈ L(1)2(R) := { g ∈ L(1)(R) : ∫ Rd |x|2 g dx <∞ }”
- Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence (2022, background): “Other remainder terms are given in [8, 35, 32] and, using weaker notions of distances, in [8, 49, 35, 37, 48] while some obstructions to stability results involving strong notions of distance are given in [50, 34].”
- Transport Inequalities for Log-concave Measures, Quantitative Forms, and Applications (2015, result): “The result obtained by Fathi, Indrei and Ledoux for the standard Gaussian measure µ = γ on Rn (a case where λ(µ) = 1) is as follows: for any probability measure ν with ∫ x dν(x) = ∫ x dγ(x) = 0, H(ν||γ)− 1 2 W 22 (γ, ν) ≥ cmin (W 21,1(γ, ν) n , W1,1(γ, ν)√ n ) where W1,1 := W‖x−y‖1 with ‖x− y‖1 := ∑”