Boundary behavior and optimal pointwise H\"older exponent of the total local time of $(1+\beta)$-stable super-Brownian motion
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 \[ h_L(\mathsf L)=h_L(\mathsf R)=1+\frac{2}{\beta} \] almost surely, where $h_L$ denotes the pointwise H\"older exponent. Thus the pointwise $\gamma$-H\"older 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.
math.PR↗