Kinetic equations, functional inequalities, and distances in the space of probability measures

This thesis concerns kinetic Fokker-Planck equations, stability of functionals inequalities, and nonlinear Dirichlet forms. Constructive convergence rates to equilibrium for kinetic equations are computed via a functional analytic framework based on weak norms of solutions. The Vlasov-Fokker-Planck equation, with the space variable confined in a torus, is analysed as a benchmark. Then, the same strategy is generalised to a wide class of kinetic Fokker-Planck models. We also consider Gagliardo-Nirenberg inequalities on the sphere, interpolating between the Poincaré and the Sobolev inequalities. We prove constructive stability results, in the strongest possible norm, with sharp exponents in the distance from optimisers. This term degenerates on a finite-dimensionai subspace requiring additional care. Our technjque combines Taylor expansions, har­monk analysis, and (non)Linear diffusion flows. We rigorously prove convergence of the Gagliardo-Nirenberg family of inequalities. for the dimension of the sphere approaching infinity, to the Gaussian Beckner ineq ual­ities. Then, we give constructive stability results for those, using nonlinear diffusion flows on the Gaussian space. Finally, we treat the Gaussian logarithmic Sobolev inequality as a limit case. We fi.nd ex.plicit stability estimates for log-concave or compactly-supported densities, thanks to the interplay between log-concavity and the Ornstein-Uhlenbeck flow. using the carré du. champ method. We contribute to the theory of nonlin­ear Dirichlet forms. We extend the normai contracùon property co the nonlinear setting. The proof adopts a new strategy, based on the approx.imation of real LipschHz functions with repeated compositions of elemen­tary piecewise linear functions.