Probability Seminar: Jieliang Hong

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.