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Solutions of Δu=4u 2 with Neumann’s conditions using the Brownian snake
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  • Published: 02 January 2004

Solutions of Δu=4u 2 with Neumann’s conditions using the Brownian snake

  • Romain Abraham1 &
  • Jean-François Delmas2 

Probability Theory and Related Fields volume 128, pages 475–516 (2004)Cite this article

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Abstract.

We consider a Brownian snake (W s ,s≥0) with underlying process a reflected Brownian motion in a bounded domain D. We construct a continuous additive functional (L s ,s≥0) of the Brownian snake which counts the time spent by the end points Ŵ s of the Brownian snake paths on ∂D. The random measure Z=∫δŴ sdL s is supported by ∂D. Then we represent the solution v of Δu=4u 2 in D with weak Neumann boundary condition φ≥0 by using exponential moment of (Z,φ) under the excursion measure of the Brownian snake. We then derive an integral equation for v. For small φ it is then possible to describe negative solution of Δu=4u 2 in D with weak Neumann boundary condition φ. In contrast to the exit measure of the Brownian snake out of D, the measure Z is more regular. In particular we show it is absolutely continuous with respect to the surface measure on ∂D for dimension 2 and 3.

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Author information

Authors and Affiliations

  1. UFR de Mathématiques et d’Informatique, Université René Descartes, 45 rue des Saints Pères, 75270, Paris Cedex 06, France

    Romain Abraham

  2. ENPC-CERMICS, 6-8 av., Blaise Pascal, Champs-sur-Marne, 77455, Marne La Vallée, France

    Jean-François Delmas

Authors
  1. Romain Abraham
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  2. Jean-François Delmas
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Correspondence to Jean-François Delmas.

Additional information

Mathematics Subject Classification (2000): 60J55, 60J80, 60H30, 60G57, 35C15, 35J65

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Abraham, R., Delmas, JF. Solutions of Δu=4u 2 with Neumann’s conditions using the Brownian snake. Probab. Theory Relat. Fields 128, 475–516 (2004). https://doi.org/10.1007/s00440-003-0302-2

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  • Received: 24 June 2002

  • Revised: 10 September 2003

  • Published: 02 January 2004

  • Issue Date: April 2004

  • DOI: https://doi.org/10.1007/s00440-003-0302-2

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Keywords or phrases

  • Super Brownian motion
  • Brownian snake
  • Exit measure
  • Neumann’s problem
  • Reflected Brownian motion
  • Semi-linear PDE
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