Limiting Distributions for a Class of Super-Brownian Motions with Spatially Dependent Branching Mechanisms

In this paper, we consider a large class of super-Brownian motions in \({\mathbb {R}}\) with spatially dependent branching mechanisms. We establish the almost sure growth rate of the mass located outside a time-dependent interval \((-\delta t,\delta t)\) for \(\delta >0\). The growth rate is given in terms of the principal eigenvalue \(\lambda _{1}\) of the Schrödinger-type operator associated with the branching mechanism. From this result, we see the existence of phase transition for the growth order at \(\delta =\sqrt{\lambda _{1}/2}\). We further show that the super-Brownian motion shifted by \(\sqrt{\lambda _{1}/2}\,t\) converges in distribution to a random measure with random density mixed by a martingale limit.