Renormalization of local times of super-Brownian motion

Abstract

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.