On the evaluation of the path integral for nonlinear diffusion processes by means of fourier series

In this paper we embark once more on the current discussion concerning the appropriate expression for the Lagrangian occurring in the action in the functional integral representation of continuous Markov processes. The equivalent differential equation representation of diffusion processes, that is the Fokker-Planck equation, will be derived using a Fourier series expansion for the path x(t) between the prepoint and the postpoint in the short time propagator. We thus allow for an arbitrary path rather than the usually considered a priori straight line. The present result solidifies earlier results from Stratonovich, Graham, Horsthemke and Bach, and ourselves.