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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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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DOI: https://doi.org/10.1007/s004400050103
- Mathematics Subject Classification (1991): Primary
- 60J45
- 60G57; Secondary
- 60J55
- 60J50
- 35J65