Summary
We study some properties of the exit measure of super Brownian motion from a smooth domainD inR d. In particular, we give precise estimates for the probability that the exit measure gives a positive mass to a small ball on the boundary. As an application, we compute the Hausdorff dimension of the support of the exit measure. In dimension 2, we prove that the exit measure is absolutely continuous with respect to the Lebesgue measure on the boundary. In connection with Dynkin's work, our results give some information on the behavior of solutions of Δu=u 2 inD, and are related to the characterization of removable singularities at the boundary. As a consequence of our estimates, we give a sufficient condition for the uniqueness of the positive solution of Δu=u 2 inD that tends to ∞ on an open subsetO of ϖD and to 0 on the complement in ϖD of the closure ofO. Our proofs use the path-valued process studied in [L2, L3].
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Abraham, R., Le Gall, JF. Sur la mesure de sortie du super mouvement brownien. Probab. Th. Rel. Fields 99, 251–275 (1994). https://doi.org/10.1007/BF01199025
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DOI: https://doi.org/10.1007/BF01199025
Mathematics Subject Classification (1991)
- 60G57
- 60G17
- 60H30
- 35J60