Characterization of G-regularity for super-Brownian motion and consequences for parabolic partial differential equations

Abstract

\def\R{\mathbb R} 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 hitting probability of a Borel set $A\subset E$ for the graph of the Brownian snake 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 ${\delta }_0$ hits immediately $A$ (that is $(0,0)$ 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 $(0,0)\in \partial Q$, we give a necessary and sufficient condition for the existence on $Q$ of a positive solution of ${\partial }_{t}u+1/2\Delta u=2u^2$ which blows up at $(0,0)$. We also give an estimation of the hitting probabilities for the support of super-Brownian motion at fixed time. We prove that if $d\ge 2$, the support of super-Brownian motion is intersection-equivalent to the range of Brownian motion.