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Fokker–Planck processes with a singular diffusion matrix are quite frequently met in Physics and Chemistry. For a long time the resulting noninvertability of the diffusion matrix has been looked as a serious obstacle for treating these Fokker–Planck equations by various powerful numerical methods of quantum and statistical mechanics. In this paper, a path-integral method is presented that takes advantage of the singularity of the diffusion matrix and allows one to solve such problems in a simple and economic way. The basic idea is to split the Fokker–Planck equation into one of a linear system and an anharmonic correction and then to employ a symmetric decomposition of the short time propagator, which is exact up to a high order in the time step. Just because of the singularity of the diffusion matrix, the factors of the resulting product formula consist of well behaved propagators. In this way one obtains a highly accurate propagation scheme, which is simultaneously fast, stable, and computationally simp...

Path integrals for Fokker–Planck dynamics with singular diffusion: Accurate factorization for the time evolution operator