Analytical results for a Fokker-Planck equation in the small noise limit

TL;DR

This paper derives analytical expressions for the first few cumulants of a nonlinear stochastic process governed by a Fokker-Planck equation in the small noise limit, revealing how noise and nonlinearity interact.

Contribution

It provides explicit formulas for cumulants using Lambert W functions, advancing understanding of nonlinear stochastic systems far from equilibrium.

Findings

Mean, variance, and covariance expressed with Lambert W functions

Analytical results valid in the small noise limit

Insights into noise and nonlinearity interplay far from equilibrium

Abstract

We present analytical results for the lowest cumulants of a stochastic process described by a Fokker-Planck equation with nonlinear drift. We show that, in the limit of small fluctuations, the mean, the variance and the covariance of the process can be expressed in compact form with the help of the Lambert W function. As an application, we discuss the interplay of noise and nonlinearity far from equilibrium.