A Path Integral Method for Coarse-Graining Noise in Stochastic Differential Equations with Multiple Time Scales

TL;DR

This paper introduces a novel path integral approach to coarse-grain noise in stochastic differential equations with multiple time scales, enabling simplified analysis of complex systems like fiber optic communications.

Contribution

The paper develops a new path integral method combined with multi-scale expansion to derive effective slow-scale stochastic equations, advancing analysis of multi-scale stochastic systems.

Findings

Method yields the same leading-order results as Fokker-Planck asymptotics for the example system.

Successfully applied to fiber optic dispersion fluctuations.

Provides a systematic framework for coarse-graining noise in multi-scale stochastic models.

Abstract

We present a new path integral method to analyze stochastically perturbed ordinary differential equations with multiple time scales. The objective of this method is to derive from the original system a new stochastic differential equation describing the system's evolution on slow time scales. For this purpose, we start from the corresponding path integral representation of the stochastic system and apply a multi-scale expansion to the associated path integral kernel of the corresponding Lagrangian. As a concrete example, we apply this expansion to a system that arises in the study of random dispersion fluctuations in dispersion-managed fiber optic communications. Moreover, we show that, for this particular example, the new path integration method yields the same result at leading order as an asymptotic expansion of the associated Fokker-Planck equation.