Analytical and Geometric Foundations and Modern Applications of Kinetic Equations and Optimal Transport

We develop a unified analytical framework linking kinetic theory, optimal transport, and entropy dissipation through the lens of hypocoercivity. Centered on the Boltzmann and Fokker–Planck equations, we analyze the emergence of macroscopic irreversibility from time-reversible dynamics via entropy methods, functional inequalities, and commutator estimates. The hypercoercivity approach provides sharp exponential convergence rates under minimal regularity, resolving degeneracies in kinetic operators through geometric control. We extend this framework to the study of hydrodynamic limits, collisional relaxation in magnetized plasmas, and the Vlasov–Poisson system for self-gravitating matter. Additionally, we explore connections with high-dimensional data analysis, where Wasserstein gradient flows, entropic regularization, and kinetic Langevin dynamics underpin modern generative and sampling algorithms. Our results highlight entropy as a structural and variational tool across both physical and algorithmic domains.