Fokker-Planck equation with variable diffusion coefficient in the Stratonovich approach

Abstract

We consider the Langevin equation with multiplicative noise term which depends on time and space. The corresponding Fokker-Planck equation in Stratonovich approach is investigated. Its formal solution is obtained for an arbitrary multiplicative noise term given by \(g(x,t)=D(x)T(t)\), and the behaviors of probability distributions, for some specific functions of \(D(x)\)% , are analyzed. In particular, for \(D(x)\sim | x| ^{-\theta /2}\), the physical solutions for the probability distribution in the Ito, Stratonovich and postpoint discretization approaches can be obtained and analyzed.