Homotopy analysis method for stochastic differential equations

TL;DR

This paper applies the homotopy analysis method to stochastic differential equations and Fokker-Planck equations, demonstrating its effectiveness and potential in quantum field theory and statistical mechanics.

Contribution

It extends the homotopy analysis method to stochastic equations, showing promising results for applications in quantum physics and statistical mechanics.

Findings

Excellent agreement with exact solutions where available

Effective for stochastic differential and Fokker-Planck equations

Potential applications in quantum field theory

Abstract

The homotopy analysis method known from its successful applications to obtain quasi-analytical approximations of solutions of ordinary and partial differential equations is applied to stochastic differential equations with Gaussian stochastic forces and to the Fokker-Planck equations. Only the simplest non-trivial examples of such equations are considered, but such that they can almost immediately be translated to those which appear in the stochastic quantization of a nonlinear scalar field theory. It has been found that the homotopy analysis method yields excellent agreement with exact results (when the latter are available) and appears to be a very promising approach in the calculations related to quantum field theory and quantum statistical mechanics.