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Abstract

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 $0