An almost sure limit theorem for super-Brownian motion

TL;DR

This paper proves an almost sure scaling limit theorem for super-Brownian motion in , extending previous results to cases where spectral assumptions fail, with applications to certain sub-domains.

Contribution

It establishes a new almost sure limit theorem for super-Brownian motion without relying on spectral assumptions, broadening the scope of previous results.

Findings

The theorem applies to super-Brownian motion in  with specific semi-linear PDEs.

Results hold even when spectral assumptions are not satisfied.

The theorem extends to some sub-domains in .

Abstract

We establish an almost sure scaling limit theorem for super-Brownian motion on $\mathbb{R}^d$ associated with the semi-linear equation $u_t = {1/2}\Delta u +\beta u-\alpha u^2$, where $\alpha$ and $\beta$ are positive constants. In this case, the spectral theoretical assumptions that required in Chen et al (2008) are not satisfied. An example is given to show that the main results also hold for some sub-domains in $\mathbb{R}^d$.