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Wiener’s test for super-Brownian motion and the Brownian snake
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  • Published: May 1997

Wiener’s test for super-Brownian motion and the Brownian snake

  • Jean-Stéphane Dhersin1 &
  • Jean-François Le Gall1 

Probability Theory and Related Fields volume 108, pages 103–129 (1997)Cite this article

Abstract.

We prove a Wiener-type criterion for super-Brownian motion and the Brownian snake.If F is a Borel subset of ℝ d and x ∈ ℝ d, we provide a necessary and sufficientcondition for super-Brownian motion started at δ x to immediately hit the set F. Equivalently, this condition is necessary and sufficient for the hitting time of F by theBrownian snake with initial point x to be 0. A key ingredient of the proof isan estimate showing that the hitting probability of F is comparable, up to multiplicative constants,to the relevant capacity of F. This estimate, which is of independent interest, refines previous results due to Perkins and Dynkin. An important role is played by additivefunctionals of the Brownian snake, which are investigated here via the potentialtheory of symmetric Markov processes. As a direct application of our probabilisticresults, we obtain a necessary and sufficient condition for the existence in a domain D of a positivesolution of the equation Δ; u = u 2 which explodes at a given point of ∂ D.

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Authors and Affiliations

  1. Laboratoire de Probabilités, Université Paris 6, 4 Place Jussieu, F-75252 Paris Cedex 05, France (e-mail: gall@ccr.jussieu.fr), , , , , , FR

    Jean-Stéphane Dhersin & Jean-François Le Gall

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  1. Jean-Stéphane Dhersin
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  2. Jean-François Le Gall
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Received: 5 January 1996 / In revised form: 30 October 1996

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Dhersin, JS., Gall, JF. Wiener’s test for super-Brownian motion and the Brownian snake. Probab Theory Relat Fields 108, 103–129 (1997). https://doi.org/10.1007/s004400050103

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  • Issue Date: May 1997

  • DOI: https://doi.org/10.1007/s004400050103

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  • Mathematics Subject Classification (1991): Primary
  • 60J45
  • 60G57; Secondary
  • 60J55
  • 60J50
  • 35J65
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