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 R with spatially dependent branching mechanisms. We establish the almost sure growth rate of the mass located outside a time-dependent interval (-δt,δt) for δ>0. The growth rate is given in terms of the principal eigenvalue λ₁ 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 δ=λ₁/2. We further show that the super-Brownian motion shifted by λ₁/2t converges in distribution to a random measure with random density mixed by a martingale limit.