{
  "id": "1305.1351",
  "version": "v2",
  "published": "2013-05-06T22:56:59.000Z",
  "updated": "2019-10-18T22:07:25.000Z",
  "title": "Exit densities of Super--Brownian motion as extreme X-harmonic functions",
  "authors": [
    "A. Deniz Sezer"
  ],
  "comment": "There is a mistake in one of the proofs",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X$ be a super-Brownian motion (SBM) defined on a domain $E\\subset R^n$ and $(X_D)$ be its exit measures indexed by sub-domains of $E$. The relationship between the equation $1/2 \\Delta u=2 u^2$ and Super-Brownian motion (SBM) is analogous to the relationship between Brownian motion and the Laplace's equation, and substantial progress has been made on the study of the solutions of this semi-linear p.d.e. exploring this analogy. An area that remains to be explored is Martin boundary theory. Martin boundary in the semi-linear case is defined as the convex set of extreme $X$-harmonic functions which are functions on the space of finite measures supported in a domain $E$ of $R^d$ and characterized by a mean value property with respect to the Super-Brownian law. So far no probabilistic construction of Martin boundary is known. In this paper, we consider a bounded smooth domain $D$, and we investigate exit densities of SBM, a certain family of $X$ harmonic functions, $H^{\\nu}$, indexed by finite measures $\\nu$ on $\\partial{D}$, These densities were first introduced by E.B. Dynkin and also identified by T.Salisbury and D. Sezer as the extended X-harmonic functions corresponding to conditioning SBM on its exit measure $X_D$ being equal to $\\nu$. $H^{\\nu}(\\mu)$ can be thought as the analogue of the Poisson kernel for Brownian motion. It is well known that Poisson kernel for a smooth domain $D$ is equivalent to the so called Martin kernel, the class of extreme harmonic functions for $D$. We show that a similar result is true for Super-Brownian motion as well, that is $H^{\\nu}$ is extreme for almost all $\\nu$ with respect to a certain measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-06T22:56:59.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2019-10-18T22:07:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "super-brownian motion",
      "extreme x-harmonic functions",
      "exit densities",
      "finite measures",
      "exit measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.1351D"
    }
  }
}