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This lecture is the fourth part of a series on Regularity Theory of Kinetic Equations with Rough Coefficients, presented by Clément Mouhot from the University of Cambridge. Explore the extension of De Giorgi (1958) and Nash (1959) theory to hypoelliptic PDEs in kinetic theory, focusing on equations with rough coefficients. The lecture examines the Kolmogorov equation (kinetic Fokker-Planck equation) with rough matrices of coefficients in kinetic diffusion, highlighting quantitative robust methods based on trajectory construction and their connections to control theory and hypocoercivity. This two-hour presentation is part of a broader series that addresses a recent active research area in PDE analysis, building upon the work of various mathematicians including Pascucci-Polidoro, Wang-Zhang, Golse-Imbert-Mouhot-Vasseur, and others who have contributed to extending Hölder regularity solutions to the hypoelliptic context.
