Exact solution of the Fokker-Planck equation for a broad class of diffusion coefficients

We consider the Langevin equation with a multiplicative noise term that depends on time and space. The corresponding Fokker-Planck equation in the Stratonovich approach is investigated. Its exact 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. We show that the asymptotic shape of the random-walk model and power-law decay obtained from other approaches can be reproduced from our solutions, by employing two simple functions for g(x,t). In particular, for D(x)∼∣x∣^−θ∕2, the physical solutions for the probability distribution in the Ito, Stratonovich, and postpoint discretization approaches can be obtained and analyzed.