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April 1999 Characterization of $G$-Regularity for Super-Brownian Motion and Consequences for Parabolic Partial Differential Equations
Jean-François Delmas, Jean-Stéphane Dhersin
Ann. Probab. 27(2): 731-750 (April 1999). DOI: 10.1214/aop/1022677384


We give a characterization of $G$-regularity for super-Brownian motion and the Brownian snake. More precisely, we define a capacity on $E = (0,\infty) \times\mathbb{R}^d$, which is not invariant by translation. We then prove that the measure of hitting a Borel set $A \subset E$ for the graph of the Brownian snake excursion starting at (0,0) is comparable, up to multiplicative constants, to its capacity. This implies that super-Brownian motion started at time 0 at the Dirac mass $\varrho_0$ hits immediately $A$ [i.e., (0,0) is $G$ -regular for $A_c$] if and only if its capacity is infinite. As a direct consequence, if $Q \subset E$ is a domain such that (0,0) $\in\varrho Q$, we give a necessary and sufficient condition for the existence on $Q$ of a positive solution of $\varrho_tu + 1/2\Delta u = 2u^2$, which blows up at (0,0). We also give an estimate of the hitting probabilities for the support of super-Brownian motion at fixed time. We prove that if $d\geq 2$, the support of super-Brownian motion is intersection-equivalent to the range of Brownian motion.


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Jean-François Delmas. Jean-Stéphane Dhersin. "Characterization of $G$-Regularity for Super-Brownian Motion and Consequences for Parabolic Partial Differential Equations." Ann. Probab. 27 (2) 731 - 750, April 1999.


Published: April 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0943.60085
MathSciNet: MR1698959
Digital Object Identifier: 10.1214/aop/1022677384

Primary: 35K60 , 60G57

Keywords: Brownian snake , capacity , G -regularity , hitting probabilities , intersection-equivalence , parabolic nonlinear PDE , Super-Brownian motion

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 2 • April 1999
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