Approximate Slow Manifolds in the Fokker-Planck Equation
Christian Kuehn, Jan-Eric Sulzbach
TL;DR
This work develops a geometric slow-manifold framework for fast-slow Fokker-Planck equations arising from multiscale SDEs by projecting onto the fast-invariant density via a Sturm–Liouville basis. It constructs an infinite-dimensional coefficient system, applies Galerkin truncation to obtain a finite fast-slow system, and proves a Fenichel-type theorem ensuring a locally attracting slow manifold with $\mathcal{O}(\varepsilon)$-accurate reduced dynamics. The approach unifies projection/Mori–Zwanzig ideas with spectral decompositions to enable rigorous dimension reduction at the PDE level, with concrete results demonstrated on a linear example using Hermite functions. The methodology provides a principled route to approximate the evolution of slow variables in stochastic multiscale settings and offers avenues for extension to non-Gaussian noise and stochastic averaging contexts, with practical implications for model reduction and multiscale analysis.
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.