Variational methods for the kinetic Fokker-Planck equation

Abstract

We develop a functional analytic approach to the study of the Kramers and kinetic Fokker-Planck equations which parallels the classical $H^1$ theory of uniformly elliptic equations. In particular, we identify a function space analogous to $H^1$ and develop a well-posedness theory for weak solutions in this space. In the case of a conservative force, we identify the weak solution as the minimizer of a uniformly convex functional. We prove new functional inequalities of Poincaré and Hörmander type and combine them with basic energy estimates (analogous to the Caccioppoli inequality) in an iteration procedure to obtain the $C^{\infty }$ regularity of weak solutions. We also use the Poincaré-type inequality to give an elementary proof of the exponential convergence to equilibrium for solutions of the kinetic Fokker-Planck equation which mirrors the classic dissipative estimate for the heat equation. Finally, we prove enhanced dissipation in a weakly collisional limit.

This is a major revision: two authors have been added, some mistakes from the previous version have been addressed, and some new results have been included (Hormander inequality now has an optimal exponent, enhanced dissipation result has been added). 53 pages

References in corpus (4)

• Quantitative stochastic homogenization and regularity theory of parabolic equations
• On explicit $L^2$-convergence rate estimate for underdamped Langevin dynamics
• On explicit $L^2$-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms
• Weighted $L^2$-contractivity of Langevin dynamics with singular potentials

Cited by in corpus (6)

• Mixing Time of Open Quantum Systems via Hypocoercivity
• Non-reversible lifts of reversible diffusion processes and relaxation times
• Regularity of kinetic Fokker-Planck equations in bounded domains
• Hypocoercivity meets lifts
• A note on asymptotics of linear dissipative kinetic equations in bounded domains
• On a kinetic Poincaré inequality and beyond