title: Exact solution of the Fokker-Planck equation for a broad class of diffusion coefficients.
abstract: 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) approximately /x/(-theta/2) , the physical solutions for the probability distribution in the Ito, Stratonovich, and postpoint discretization approaches can be obtained and analyzed.
authors.affiliation: Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo, 5790, 87020-900, Maringá-Paraná, Brazil.
authors.forename: Kwok Sau
authors.initials: KS
authors.lastname: Fa