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Some properties of the exit measure for super Brownian motion
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  • Published: January 2002

Some properties of the exit measure for super Brownian motion

  • Romain Abraham1 &
  • Jean-François Delmas2 

Probability Theory and Related Fields volume 122, pages 71–107 (2002)Cite this article

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  • 1 Citations

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Abstract.

 We consider the exit measure of super Brownian motion with a stable branching mechanism of a smooth domain D of ℝd. We derive lower bounds for the hitting probability of small balls for the exit measure and upper bounds in the critical dimension. This completes results given by Sheu [22] and generalizes the results of Abraham and Le Gall [2]. Because of the links between exits measure and partial differential equations, those results imply bounds on solutions of elliptic semi-linear PDE. We also give the Hausdorff dimension of the support of the exit measure and show it is totally disconnected in high dimension. Eventually we prove the exit measure is singular with respect to the surface measure on ∂D in the critical dimension. Our main tool is the subordinated Brownian snake introduced by Bertoin, Le Gall and Le Jan [4].

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Authors and Affiliations

  1. UFR de Mathématiques et d’Informatique, Université René Descartes, 45 rue des Saints Pères, 75270 Paris Cedex 06, France. e-mail: abraham@math-info.univ-paris5.fr, , , , , , FR

    Romain Abraham

  2. ENPC-CERMICS, 6-8 av. Blaise Pascal, Champs-sur-Marne, 77455 Marne La Vallée, France. e-mail: delmas@cermics.enpc.fr, , , , , , FR

    Jean-François Delmas

Authors
  1. Romain Abraham
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  2. Jean-François Delmas
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Received: 6 December 1999 / Revised version: 9 June 2000 / Published online: 11 December 2001

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Abraham, R., Delmas, JF. Some properties of the exit measure for super Brownian motion. Probab Theory Relat Fields 122, 71–107 (2002). https://doi.org/10.1007/s004400100181

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  • Issue Date: January 2002

  • DOI: https://doi.org/10.1007/s004400100181

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Keywords

  • Differential Equation
  • Gall
  • Partial Differential Equation
  • Lower Bound
  • Brownian Motion
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