{
  "id": "1611.04869",
  "version": "v1",
  "published": "2016-11-15T14:49:50.000Z",
  "updated": "2016-11-15T14:49:50.000Z",
  "title": "Spectral theory for random Poincaré maps",
  "authors": [
    "Manon Baudel",
    "Nils Berglund"
  ],
  "comment": "58 pages, 5 figures",
  "categories": [
    "math.PR",
    "math.DS"
  ],
  "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 discovred detailed-balance property satisfied by committor functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-15T14:49:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J60",
      "60J35",
      "34F05",
      "45B05"
    ],
    "keywords": [
      "spectral theory",
      "eigenvalues",
      "periodic orbits",
      "committor functions",
      "adding weak gaussian white noise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}