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

Yan-Xia Ren () and Ting Yang ()
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Yan-Xia Ren: Peking University
Ting Yang: Beijing Institute of Technology

Journal of Theoretical Probability, 2024, vol. 37, issue 3, 2457-2507

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

Keywords: Super-Brownian motion; Spatially dependent branching mechanism; Growth rate; Convergence in distribution; Primary 60J68; 60G57; Secondary 60F15; 60F05 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-023-01304-2

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