Gaussian fluctuation for spatial average of super-Brownian motion

TL;DR

This paper proves that the normalized spatial average of a one-dimensional super-Brownian motion converges to a Brownian sheet as the spatial scale grows, using the Laplace functional to analyze fluctuations.

Contribution

It establishes a Gaussian fluctuation limit for the spatial average of super-Brownian motion, extending understanding of its asymptotic behavior.

Findings

Normalized spatial integral converges to Brownian sheet

Joint convergence in (t, x) variables

Uses Laplace functional for proof

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 $N^{-1/2}\int_0^{xN}[u(t\,, z)-1 ]\rm{d} z$ converges jointly in $(t, x)$ to Brownian sheet in distribution.