Franz Achleitner^[1], Anton Arnold^[2] and Dominik Stürzer^[3],

Large-time behavior in non-symmetric Fokker-Planck equations

Pages: 1-68
Received: 20 March 2015   
Accepted : 6 June 2015
Mathematics Subject Classification (2010): Primary 35Q84, 35H10, 35B20; Secondary 35K10, 35B40, 47D07, 35P99, 47D06.
Keywords: Fokker-Planck equation, hypocoercivity, entropy method, large-time behavior, spectral gap, sharp decay rate, non-local perturbation, spectral analysis, exponential stability.
Authors addresses:
[1],[2],[3] : Vienna University of Technology, Wiedner Hauptstrasse 8, Vienna, 1040, Austria.

This research was partially supported by the FWF-doctoral school "Dissipation and dispersion in nonlinear partial differential equations" and INDAM - GNFM from Italy. One author (AA) is grateful to J. Schöberl for very helpful discussions.

Abstract: We consider three classes of linear non-symmetric Fokker-Planck equations having a unique steady state and establish exponential convergence of solutions towards the steady state with explicit (estimates of) decay rates. First, "hypocoercive" Fokker-Planck equations are degenerate parabolic equations such that the entropy method to study large-time behavior of solutions has to be modified. We review a recent modified entropy method (for non-symmetric Fokker-Planck equations with drift terms that are linear in the position variable). Second, kinetic Fokker-Planck equations with non-quadratic potentials are another example of non-symmetric Fokker-Planck equations. Their drift term is nonlinear in the position variable. In case of potentials with bounded second-order derivatives, the modified entropy method allows to prove exponential convergence of solutions to the steady state. In this application of the modified entropy method symmetric positive definite matrices solving a matrix inequality are needed. We determine all such matrices achieving the optimal decay rate in the modified entropy method. In this way we prove the optimality of previous results. Third, we discuss the spectral properties of Fokker-Planck operators perturbed with convolution operators. For the corresponding Fokker-Planck equation we show existence and uniqueness of a stationary solution. Then, exponential convergence of all solutions towards the stationary solution is proven with a uniform rate.

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