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Exponential convergence to equilibrium for kinetic Fokker-Planck equations. (English) Zbl 1254.35220
Summary: A class of linear kinetic Fokker-Planck equations with a non-trivial diffusion matrix and with periodic boundary conditions in the spatial variable is considered. After formulating the problem in a geometric setting, the question of the rate of convergence to equilibrium is studied within the formalism of differential calculus on Riemannian manifolds. Under explicit geometric assumptions on the velocity field, the energy function and the diffusion matrix, it is shown that global regular solutions converge in time to equilibrium with exponential rate. The result is proved by estimating the time derivative of a modified entropy functional, as recently proposed by C. Villani [Mem. Am. Math. Soc. 950, 1–141 (2009; Zbl 1197.35004)]. For spatially homogeneous solutions the assumptions of the main theorem reduce to the curvature bound condition for the validity of logarithmic Sobolev inequalities discovered by D. Bakry and M. Emery [C. R. Acad. Sci., Paris, Sér. I 299, 775–778 (1984; Zbl 0563.60068)]. The result applies to the relativistic Fokker-Planck equation in the low temperature regime, for which exponential trend to equilibrium was previously unknown.

35Q84 Fokker-Planck equations
35B40 Asymptotic behavior of solutions to PDEs
58J65 Diffusion processes and stochastic analysis on manifolds
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