Stochastic integral representation and regularity of the density for the Exit measure of super-Brownian motion

Details Stochastic integral representation and regularity of the density for the Exit measure of super-Brownian motion Stochastic integral representation and regularity of the density for the Exit measure of super-Brownian motion

2005-03-25

arxiv.org/abs/math/0503599v1

Annals of Probability 2005, Vol. 33, No. 1, 194-222

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117904000000612 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117904000000612

This paper studies the regularity properties of the density of the exit measure for super-Brownian motion with (1+\beta)-stable branching mechanism. It establishes the continuity of the density in dimension d=2 and the unboundedness of the density in all other dimensions where the density exists. An alternative description of the exit measure and its density is also given via a stochastic integral representation. Results are applied to the probabilistic representation of nonnegative solutions of the partial differential equation \Delta u=u^{1+\beta}.

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