SPECTRAL THEORY FOR RANDOM POINCARE MAPS

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 Poincare map, which encodes the metastable behavior of the system. We show that this process admits exactly N eigenvalues which are exponentially close to 1, and we provide expressions for these eigenvalues and their left and right eigenfunctions in terms of committor functions of neighborhoods of periodic orbits. The eigenvalues and eigenfunctions are well-approximated by principal eigenvalues and quasi-stationary distributions of processes killed upon hitting some of these neighborhoods. 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 satis fi ed by committor functions.