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We consider the kinetic Fokker–Planck equation with a class of general force. We prove the existence and uniqueness of a positive normalized equilibrium (in the case of a general force) and establish some exponential rate of convergence to the equilibrium (and the rate can be explicitly computed). Our results improve results about classical force to general force case. Our result also improves the rate of convergence for the Fitzhugh–Nagumo equation from non-quantitative to quantitative explicit rate.

The kinetic Fokker–Planck equation with general force