{
  "id": "2110.01233",
  "version": "v2",
  "published": "2021-10-04T07:53:27.000Z",
  "updated": "2023-06-26T11:47:14.000Z",
  "title": "Poisson-Orlicz norm and infinite Ergodic Theory",
  "authors": [
    "Emmanuel Roy"
  ],
  "categories": [
    "math.DS",
    "math.PR"
  ],
  "abstract": "Urbanik's theorem for a Poisson process on an infinite measure space (X, A, $\\mu$) relates integrability of stochastic integrals to a particular Orlicz function space L$\\Phi$ ($\\mu$) on which the L1-norm of the Poisson process induces a norm (called Poisson-Orlicz in the sequel) that is shown to be equivalent to the classical gauge and Orlicz norms.We obtain a full characterization of stochastic integrals using difference operators that, together with a simple duality argument, allows to derive Urbanik's theorem as well as an optimal inequality between the Orlicz and the Poisson-Orlicz norm.In a second part, we show that the Poisson-Orlicz norm plays a role in infinite Ergodic Theory where it is seen as an alternative to the L1-norm to identify several dynamical invariants that the latter fails to identify. We also show that, whereas the L1-norm fully characterizes exact endomorphisms (Lin's theorem), Poisson-Orlicz norm fully characterizes remotely infinite endomorphisms.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-06-26T11:47:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "infinite ergodic theory",
      "poisson-orlicz norm",
      "characterizes remotely infinite endomorphisms",
      "fully characterizes exact endomorphisms",
      "fully characterizes remotely infinite"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}