Abstract
We prove a lemma concerning the behavior of the historical paths of super-Brownian motion near their endpoint. We use this lemma to estimate the hitting probability of a small disk for two-dimensional super-Brownian motion, thus complementing results due to Dawson, Iscoe and Perkins [3] in higher dimensions. These estimates are related to the large time behavior of the solutions of a semilinear parabolic equation, which has been investigated by several authors using analytic methods.
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References
R. Abraham and J.F. Le Gall, Sur la mesure de sortie du super-mouvement brownien. Preprint (1993).
M.T. Barlow and E.A. Perkins, Historical filtrations. Preprint (1992).
D.A. Dawson, I. Iscoe and E.A. Perkins, Super-Brownian motion: Path properties and hitting probabilities, Probab. Th. Rel. Fields 83, 135–206 (1989).
D.A. Dawson and E.A. Perkins, Historical processes, Memoirs Amer. Math. Soc. 454 (1991).
E.B. Dynkin, Branching particle systems and superprocesses, Ann. Probab. 19, 1157–1194 (1991).
E.B. Dynkin, Path processes and historical superprocesses, Probab. Th. Rel. Fields 90, 1–36 (1991).
E.B. Dynkin, Superprocesses and parabolic nonlinear differential equations, Ann. Probab. 20, 942–962 (1992).
V.A. Galaktionov, S.P. Kurdyumov and A.A. Samarskii, On asymptotic “eigenfunctions” of the Cauchy problem for a nonlinear parabolic equation. English translation in: Math. USSR Sbornik 54, 421–455 (1986).
A. Gmira and L. Véron, Large time behaviour of the solutions of a semilinear parabolic equation in ℝN, J. Differential Equations 53, 258–276 (1984).
J.F. Le Gall, Brownian excursions, trees and measure-valued branching processes, Ann. Probab. 19, 1399–1439 (1991).
J.F. Le Gall, A class of path-valued Markov processes and its applications to superprocesses, Probab. Th. Rel. Fields 95, 25–46 (1993).
J.F. Le Gall, A path-valued Markov process and its connections with partial differential equations, Proceedings First European Congress of Mathematics, Birkhäuser, to appear.
J.F. Le Gall and E.A. Perkins, The Hausdorff measure of the support of two-dimensional super-Brownian motion. In preparation.
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© 1994 Springer Science+Business Media New York
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Le Gall, JF. (1994). A Lemma on Super-Brownian Motion with Some Applications. In: Freidlin, M.I. (eds) The Dynkin Festschrift. Progress in Probability, vol 34. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0279-0_14
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DOI: https://doi.org/10.1007/978-1-4612-0279-0_14
Publisher Name: Birkhäuser, Boston, MA
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Online ISBN: 978-1-4612-0279-0
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