A structure preserving scheme for the Kolmogorov–Fokker–Planck equation

In this paper we introduce a numerical scheme which preserves the behavior of solutions to the Kolmogorov Equation as time tends to infinity. The method presented is based on a self-similar change of variables technique to transform the Kolmogorov Equation into a new form, such that the problem of designing structure preserving schemes, for the original equation, amounts to building a standard scheme for the transformed equation. This transformation also has the added benefit of allowing for an exact operator splitting scheme, whereas in the original form a standard operator splitting was only second-order. Finally, we verify the preservation of long time behavior through numerical simulations.