**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

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