Approximate slow manifolds in the Fokker-Planck equation

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.