On the boundary local time measure of super-Brownian motion

Abstract

In [9] the Hausdorff dimension, $d_{f}$, of $\partial \mathcal {R}$, the topological boundary of the range of super-Brownian motion for dimensions $d=2,3$ was found; $d_{f}=4-2\sqrt {2}$ if $d=2$, and $d_{f}=(9-\sqrt {17})/2$ if $d=3$. We will refine these dimension estimates in a number of ways.

If $L^{x}$ is the total occupation local time of $d$-dimensional super-Brownian motion, $X$, for $d=2$ and $d=3$, we construct a random measure $\mathcal {L}$, called the boundary local time measure, as a rescaling of $L^{x} e^{-\lambda L^{x}} dx$ as $\lambda \to \infty $, thus confirming a conjecture of [19] and further show that the support of $\mathcal {L}$ equals $\partial \mathcal {R}$. This latter result uses a second construction of a boundary local time $\widetilde {\mathcal {L}}$ given in terms of exit measures and we prove that $\widetilde {\mathcal {L}}=c\mathcal {L}$ a.s. for some constant $c>0$. We derive reasonably explicit first and second moment measures for $\mathcal {L}$ in terms of negative dimensional Bessel processes and use them with the energy method to give a more direct proof of the lower bound of the Hausdorff dimension of $\partial \mathcal {R}$ in [9]. The construction requires a refinement of the $L^{2}$ upper bounds in [19] and [9] to exact $L^{2}$ asymptotics. The methods also refine the left tail bounds for $L^{x}$ in [19] to exact asymptotics. We conjecture that the $d_{f}$-dimensional Minkowski content of $\partial \mathcal {R}$ is equal to the total mass of the boundary local time $\mathcal {L}$ up to some constant.