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) approximately /x/(-theta/2) , the physical solutions for the probability distribution in the Ito, Stratonovich, and postpoint discretization approaches can be obtained and analyzed.
Fa
Kwok Sau
KS
Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo, 5790, 87020-900, Maringá-Paraná, Brazil.