A boundary local time for one-dimensional super-Brownian motion and applications

Abstract

For a one-dimensional super-Brownian motion with density $X(t,x)$, we construct a random measure $L_{t}$ called the boundary local time which is supported on $BZ_{t} := \partial \{x:X(t,x) = 0\}$, thus confirming a conjecture of Mueller, Mytnik and Perkins [13]. $L_{t}$ is analogous to the local time at $0$ of solutions to an SDE. We establish first and second moment formulas for $L_{t}$, some basic properties, and a representation in terms of a cluster decomposition. Via the moment measures and the energy method we give a more direct proof that $\text{dim} (BZ_{t}) = 2-2\lambda _{0}> 0$ with positive probability, a recent result of Mueller, Mytnik and Perkins [13], where $-\lambda _{0}$ is the lead eigenvalue of a killed Ornstein-Uhlenbeck operator that characterizes the left tail of $X(t,x)$. In a companion work [6], the author and Perkins use the boundary local time and some of its properties proved here to show that $\text{dim} (BZ_{t}) = 2-2\lambda _{0}$ a.s. on $\{X_{t}(\mathbb{R} ) > 0 \}$.