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

Author

Listed:
  • Yan-Xia Ren

    (Peking University)

  • Ting Yang

    (Beijing Institute of Technology)

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.

Suggested Citation

  • Yan-Xia Ren & Ting Yang, 2024. "Limiting Distributions for a Class of Super-Brownian Motions with Spatially Dependent Branching Mechanisms," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2457-2507, September.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:3:d:10.1007_s10959-023-01304-2
    DOI: 10.1007/s10959-023-01304-2
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    References listed on IDEAS

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    1. Palau, Sandra & Yang, Ting, 2020. "Law of large numbers for supercritical superprocesses with non-local branching," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 1074-1102.
    2. Thomas Madaule, 2017. "Convergence in Law for the Branching Random Walk Seen from Its Tip," Journal of Theoretical Probability, Springer, vol. 30(1), pages 27-63, March.
    3. Ren, Yan-Xia & Song, Renming & Zhang, Rui, 2021. "The extremal process of super-Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 137(C), pages 1-34.
    4. Zhen-Qing Chen & Yan-Xia Ren & Ting Yang, 2017. "Law of Large Numbers for Branching Symmetric Hunt Processes with Measure-Valued Branching Rates," Journal of Theoretical Probability, Springer, vol. 30(3), pages 898-931, September.
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