Boundary behavior and optimal pointwise Hölder exponent of the total local time of $(1+\beta )$-stable super-Brownian motion

Abstract

Let $L^x$ be the total local time of one-dimensional super-Brownian motion with $(1+\beta )$-stable branching mechanism, $0<\beta <1$. We prove that $\{x\in \mathbb{R}:L^x>0\}$ is almost surely a bounded open interval $(\mathsf{L},\mathsf{R})$ and that \[ h_L(\mathsf L)=h_L(\mathsf R)=1+\frac{2}β \] almost surely, where $h_L$ denotes the pointwise Hölder exponent. Thus the pointwise $\gamma$-Hölder condition holds at both endpoints for every $\gamma <1+2/\beta$ and fails for every $\gamma >1+2/\beta$. The same conclusions hold under the canonical excursion measure.