Characterization of $G$-Regularity for Super-Brownian Motion and Consequences for Parabolic Partial Differential Equations

Abstract

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.