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 $\psi(x)=((2\pi^2)^{-1} \log (1/|x|))^{1/2}$ 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 $x\to 0$. We also consider general initial conditions and
get some 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.