A homotopic Galerkin approach to the solution of the Fokker-Planck-Kolmogorov equation

In this paper, we present a homotopic Galerkin approach to the solution of the Fokker-Planck-Kolmogorov equation. We argue that the ideal Hilbert space to approximate the exact solution, psi* is the space L₂(dPsi*) where dPsi* is the probability measure induced on R_n by the solution psi* itself. We show that given an initial approximation of the exact solution which is sufficiently close to the exact solution, we can obtain the exact solution iteratively by propagating the solution over sufficiently small time intervals using the Galerkin projection method. Further, we show that given a family of dynamical systems, D_P, indexed by the homotopy parameter p isin [0,1], where the dynamical system corresponding to p = 0 is a system whose associated Fokker-Planck equation can be solved and p = 1 is the dynamical system of interest, and such that the associated solutions to the corresponding Fokker-Planck equations can be changed smoothly by slowly varying the homotopy parameter, the exact solution can be obtained recursively, using the Galerkin projection method, starting with the solution to the Fokker-Planck equation associated with the dynamical system D₀