{
  "id": "1405.4416",
  "version": "v1",
  "published": "2014-05-17T16:55:26.000Z",
  "updated": "2014-05-17T16:55:26.000Z",
  "title": "Stochastic analysis for Poisson processes",
  "authors": [
    "Günter Last"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "This survey is a preliminary version of a chapter of the forthcoming book \"Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-It\\^o Chaos Expansions and Stochastic Geometry\" edited by Giovanni Peccati and Matthias Reitzner. The paper develops some basic theory for the stochastic analysis of Poisson process on a general $\\sigma$-finite measure space. After giving some fundamental definitions and properties (as the multivariate Mecke equation) the paper presents the Fock space representation of square-integrable functions of a Poisson process in terms of iterated difference operators. This is followed by the introduction of multivariate stochastic Wiener-It\\^o integrals and the discussion of their basic properties. The paper then proceeds with proving the chaos expansion of square-integrable Poisson functionals, and defining and discussing Malliavin operators. Further topics are products of Wiener-It\\^o integrals and Mehler's formula for the inverse of the Ornstein-Uhlenbeck generator based on a dynamic thinning procedure. The survey concludes with covariance identities, the Poincar\\'e inequality and the FKG-inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-17T16:55:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G55",
      "60H07"
    ],
    "keywords": [
      "stochastic analysis",
      "poisson processes",
      "chaos expansion",
      "finite measure space",
      "multivariate mecke equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.4416L"
    }
  }
}