Spectral theory for random Poincaré maps

Manon Baudel, Nils Berglund
SIAM J. Math. Analysis 49(6): 4319-4375 (2017)

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 Poincaré 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 discovred detailed-balance property satisfied by committor functions.

Mathematical Subject Classification: 60J60, 60J35 (primary), 34F05, 45B05 (secondary).

Keywords and phrases: Stochastic differential equation, periodic orbit, return map, random Poincaré map, metastability, quasistationary distributions, Doob h-transform, spectral theory, Fredholm theory, stochastic exit problem.

 

Journal Homepage Published article: 10.1137/16M1103816 MR3719020 Zbl1374.60149

PDF file (828 Kb) hal-01397184 arXiv/1611.04869