Renormalization of local times of super-Brownian motion

For the local time $L_t^x$ of super-Brownian motion $X$ starting from $\delta_0$, we study its asymptotic behavior as $x\to 0$. In $d=3$, we find a normalization $\psi(x)=(1/(2\pi^2) \log (1/|x|))^{1/2}$ such that $(L_t^x-1/(2\pi|x|))/\psi(x)$ converges in distribution to standard normal as $x\to 0$. In $d=2$, we show that $L_t^x-(1/\pi)\log (1/|x|)$ converges a.s. as $x\to 0$. We also consider general initial conditions and get similar renormalization results. The behavior of the local time allows us to derive a second order term in the asymptotic behavior of a related semilinear elliptic equation.