The local time and boundary of super-Brownian motion

We study the local time of super-Brownian motion and the topological boundary of the range of super-Brownian motion in dimensions $d\leq 3$. For the local time $L_t^x$ of super-Brownian motion $X$ starting from $\delta_0$, we first study its asymptotic behaviour as $x\to 0$. In $d=3$, we find a normalization $\psi(x)=((2\pi^2)^{-1} \log (1/|x|))^{1/2}$ such that $(L_t^x-(2\pi|x|)^{-1})/\psi(x)$ converges in distribution to standard normal as $x\to 0$. In $d=2$, we show that $L_t^x-\pi^{-1} \log (1/|x|)$ converges a.s. as $x\to 0$. Next we study the local behaviour of the local time when it is small but positive. In $d=1$, we show that it is locally $\gamma$-H\"older continuous near the boundary if $03$. Let $\partial\mathcal{R}$ be the topological boundary of the range of super-Brownian motion and dim denotes Hausdorff dimension, then with probability one, for any open set $U$, $U\cap\partial\mathcal{R}\neq \varnothing$ implies \[ \operatorname{dim}(U\cap\partial\mathcal{R})=d_f:= \begin{cases} 4-2\sqrt2\approx1.17&\text{if }d=2, \\ \frac{9-\sqrt{17}}{2}\approx2.44&\text{if }d=3. \end{cases} \] These dimension estimates are also refined in a number of ways. In $d=2$ and $d=3$, we construct a random measure $\mathcal{L}$, called the boundary local time measure, whose support equals $\partial\mathcal{R}$. It is constructed as a rescaled limit of the total local time $L^x_\infty$ where mass becomes concentrated at points $x$ where $L^x_\infty$ is small but positive. It is conjectured that the $d_f$-dimensional Minkowski content of $\partial\mathcal{R}$ is equal to the total mass of the boundary local time $\mathcal{L}$ up to some constant.Science, Faculty ofMathematics, Department ofGraduat