On the boundary of the zero set of super-Brownian motion and its local time

TL;DR

This paper determines the Hausdorff dimension of the boundary of the support of one-dimensional super-Brownian motion, confirming a conjecture and providing new insights into boundary local time and support behavior.

Contribution

It proves the almost sure Hausdorff dimension of the boundary of super-Brownian motion's support, linking it to the lead eigenvalue of a killed Ornstein-Uhlenbeck process, and introduces new properties of boundary local time.

Findings

Hausdorff dimension of boundary is 2-2λ₀, approximately 0.224

Confirmed conjecture of Mueller, Mytnik, and Perkins with almost sure results

Derived new properties of boundary local time and support behavior near the edge

Abstract

If $X(t,x)$ is the density of one-dimensional super-Brownian motion, we prove that $\text{dim}(\partial\{x:X(t,x)>0\})=2-2\lambda_0\in(0,1)$ a.s. on $\{X_t\neq 0\}$, where $-\lambda_0\in(-1,-1/2)$ is the lead eigenvalue of a killed Ornstein-Uhlenbeck process. This confirms a conjecture of Mueller, Mytnik and Perkins who proved the above with positive probability. To establish this result we derive some new basic properties of a recently introduced boundary local time and analyze the behaviour of $X(t,\cdot)$ near the upper edge of its support. Numerical estimates of $\lambda_0$ suggest that the above Hausdorff dimension is approximately $.224$.