Spectral theory for random Poincaré maps

Manon Baudel (1, 2) , Nils Berglund (1, 2)

1 FDP - Fédération de recherche Denis Poisson
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

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 discovered detailed-balance property satisfied by committor functions.

Mots clés: Stochastic differential equation; periodic orbit; return map; random Poincaré map; metastability; quasistationary distribution; Doob h-transform; spectral theory; Fredholm theory; stochastic exit problem

Domaines: Probabilités [math.PR]; Systèmes dynamiques [math.DS]

Manon Baudel, Nils Berglund. Spectral theory for random Poincaré maps. SIAM Journal on Mathematical Analysis, 2017, 49 (6), pp.4319-4375. ⟨10.1137/16M1103816⟩. ⟨hal-01397184v2⟩