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

Source abstract

Let $L^x$ be the total local time of one-dimensional super-Brownian motion with $(1+β)$-stable branching mechanism, $0 0\}$ 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 $γ$-Hölder condition holds at both endpoints for every $γ 1+2/β$. The same conclusions hold under the canonical excursion measure.