On the kinetic Fokker--Planck equation in curved geometry

Abstract

These notes are devoted to the kinetic Fokker--Planck equation on the tangent bundle of a compact Riemannian manifold, with motivations coming from relativistic kinetic theory of gases (\`a la Debbasch) and geometric analysis (\`a la Bismut). They are developed in particular from collaboration with Fabrice Debbasch and Yann Ollivier. The presentation is mostly self-contained, intended for readers without much prior familiarity with geometric analysis. Once the kinetic equations are reviewed, I expand the basic toolbox from functional and spectral analysis. Then the main focus is on global hypoelliptic $C^\infty$ regularization and hypocoercive exponential equilibration, both of which are addressed both in $L^2$ and $L^1$ settings. Various types of interpolation inequalities, spectral gap and entropic inequalities are established and used. Several existing results are simplified and unified, along with new ones. Some directions of research are mentioned.