Spectral theory for random Poincar\'e maps

TL;DR

This paper analyzes the spectral properties of a Markov chain derived from stochastic differential equations with multiple stable periodic orbits, revealing how eigenvalues and eigenfunctions relate to metastable behavior.

Contribution

It introduces a spectral analysis framework for the random Poincaré map, connecting eigenvalues and eigenfunctions to committor functions and metastability in stochastic systems.

Findings

Exactly N eigenvalues close to 1 for N stable orbits

Explicit expressions for eigenvalues and eigenfunctions

Approximation by quasistationary distributions

Abstract

We consider stochastic differential equations, obtained by adding weak Gaussian white noise to ordinary differential equations admitting $N$ asymptotically stable periodic orbits. We construct a discrete-time, continuous-space Markov chain, called a random Poincar\'e map, which encodes the metastable behaviour of the system. We show that this process admits exactly $N$ eigenvalues which are exponentially close to $1$, and provide expressions for these eigenvalues and their left and right eigenfunctions in terms of committor functions of neighbourhoods of periodic orbits. The eigenvalues and eigenfunctions are well-approximated by principal eigenvalues and quasistationary distributions of processes killed upon hitting some of these neighbourhoods. The proofs rely on Feynman--Kac-type representation formulas for eigenfunctions, Doob's $h$-transform, spectral theory of compact operators, and a recently discovered detailed-balance property satisfied by committor functions.