Gradient flow structure for McKean-Vlasov equations on discrete spaces
Matthias Erbar Max Fathi Vaios Laschos André Schlichting
In this work, we show that a family of non-linear mean-field equations on discrete spaces can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that this gradient flow structure arises as the limit of the gradient flow structures of a natural sequence of $N$-particle dynamics, as $N$ goes to infinity.
keywords: McKean-Vlasov dynamics nonlinear Markov chains mean-field limit weakly interacting particles systems Gradient flow structure evolutionary Gamma convergence transportation metric.
Quantitative logarithmic Sobolev inequalities and stability estimates
Max Fathi Emanuel Indrei Michel Ledoux
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.
keywords: deficit estimates optimal transport theory semigroup theory. transportation inequalities Logarithmic Sobolev inequalities

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