Théorie spectrale pour des applications de Poincaré aléatoires aléatoires

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 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. We also provide a bound for the remaining part of the spectrum. The eigenvalues that are exponentially close to 1 and the right and left eigenfunctions are well-approximated by principal eigenvalues, quasistationary distributions, and principal right eigenfunctions of processes killed upon hitting some of these neighbourhoods. Each eigenvalue that is exponentially close to 1 is also related to the mean exit time from some metastable neighborhood of the periodic orbits. 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.
Complete list of metadatas

Cited literature [90 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-01866385
Contributor : Manon Baudel <>
Submitted on : Monday, September 3, 2018 - 12:29:34 PM
Last modification on : Tuesday, September 11, 2018 - 1:21:36 AM
Long-term archiving on: Tuesday, December 4, 2018 - 5:13:17 PM

File

manon-baudel-These.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : tel-01866385, version 1

Collections

Citation

Manon Baudel. Théorie spectrale pour des applications de Poincaré aléatoires aléatoires. Probabilités [math.PR]. Université d'Orléans, 2017. Français. ⟨tel-01866385⟩

Share

Metrics

Record views

159

Files downloads

183