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

TL;DR

This paper derives and analyzes the Fokker-Planck equation with variable diffusion in the Stratonovich framework, providing formal solutions and exploring probability distribution behaviors for specific multiplicative noise functions.

Contribution

It presents a formal solution to the Fokker-Planck equation with time- and space-dependent diffusion in the Stratonovich approach, including analysis for particular noise functions.

Findings

Formal solution for arbitrary multiplicative noise g(x,t)=D(x)T(t).

Analysis of probability distributions for specific D(x) functions.

Comparison of solutions across Ito, Stratonovich, and postpoint discretizations.

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.