Gaussian fluctuation for spatial average of super-Brownian motion

Abstract

Let $\{u(t,x){\}}_{(t,x)\in {\mathbb{R}}_{+}\times \mathbb{R}}$ be the density of one-dimensional super-Brownian motion starting from Lebesgue measure. Using the Laplace functional of super-Brownian motion, we prove that as $N\to \infty$, the normalized spatial integral $N^{-1/2}{\int }_0^{xN}[u(t,z)-1]\mathrm{d}\mathrm{z}$ converges jointly in $(t,x)$ to Brownian sheet in distribution.