Approximate Slow Manifolds in the Fokker-Planck Equation

TL;DR

This paper develops a method to approximate slow manifolds in the Fokker-Planck PDE for multiscale SDEs, enabling reduction of complex dynamics to lower-dimensional models using spectral and geometric singular perturbation techniques.

Contribution

It introduces a novel approach combining spectral projection and geometric singular perturbation theory to approximate slow manifolds in infinite-dimensional PDEs.

Findings

Successful approximation of slow manifolds in Fokker-Planck equations

Reduction of multiscale SDE dynamics to lower-dimensional models

Integration of spectral methods with geometric singular perturbation theory

Abstract

In this paper we study the dynamics of a fast-slow Fokker-Planck partial differential equation (PDE) viewed as the evolution equation for the density of a multiscale planar stochastic differential equation (SDE). Our key focus is on the existence of a slow manifold on the PDE level, which is a crucial tool from the geometric singular perturbation theory allowing the reduction of the system to a lower dimensional slowly evolving equation. In particular, we use a projection approach based upon a Sturm- Liouville eigenbasis to convert the Fokker-Planck PDE to an infinite system of PDEs that can be truncated/approximated to any order. Based upon this truncation, we can employ the recently developed theory for geometric singular perturbation theory for slow manifolds for infinite-dimensional evolution equations. This strategy presents a new perspective on the dynamics of multiple time-scale SDEs as it combines ideas from several previously disjoint reduction methods.