A Boundary Local Time For One-Dimensional Super-Brownian Motion And Applications

TL;DR

This paper constructs and analyzes a boundary local time for one-dimensional super-Brownian motion, confirming a conjecture and providing tools to understand the boundary's fractal dimension and structure.

Contribution

It introduces the boundary local time for super-Brownian motion, proves moment formulas, and establishes the boundary's fractal dimension, confirming a prior conjecture.

Findings

Construction of boundary local time supported on the zero boundary set.

First and second moment formulas for the boundary local time.

Proof that the boundary's Hausdorff dimension is positive and equals 2-2λ₀.

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 $\partial \{x:X(t,x) = 0\} =: BZ_t$, thus confirming a conjecture of Mueller, Mytnik and Perkins (2017). $L_t$ 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 (2017), 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, 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 $\{X_t(\mathbb{R}) > 0 \}$.