Limiting Distributions for a Class of Super-Brownian Motions with Spatially Dependent Branching Mechanisms
Springer Science and Business Media LLC - Trang 1-51 - 2023
Tóm tắt
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.
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