Perturbation theory for a stochastic process with Ornstein-Uhlenbeck noise

TL;DR

This paper develops a perturbation theory for analyzing a one-dimensional stochastic process driven by Ornstein-Uhlenbeck noise, using a power series expansion of the Fokker-Planck equation with algebraic operator methods.

Contribution

It introduces a novel perturbation approach for solving the Fokker-Planck equation near an attractive fixed point with Ornstein-Uhlenbeck noise.

Findings

Exact coefficients obtained via algebraic operator methods

Power series expansion valid near fixed points

Enhanced analytical understanding of correlated noise effects

Abstract

The Ornstein-Uhlenbeck process may be used to generate a noise signal with a finite correlation time. If a one-dimensional stochastic process is driven by such a noise source, it may be analysed by solving a Fokker-Planck equation in two dimensions. In the case of motion in the vicinity of an attractive fixed point, it is shown how the solution of this equation can be developed as a power series. The coefficients are determined exactly by using algebraic properties of a system of annihilation and creation operators.