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The extremal process of super-Brownian motion

Author

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  • Ren, Yan-Xia
  • Song, Renming
  • Zhang, Rui

Abstract

In this paper, we establish limit theorems for the supremum of the support, denoted by Mt, of a supercritical super-Brownian motion {Xt,t≥0} on R. We prove that there exists an m(t) such that (Xt−m(t),Mt−m(t)) converges in law, and give some large deviation results for Mt as t→∞. We also prove that the limit of the extremal process Et≔Xt−m(t) is a Poisson random measure with exponential intensity in which each atom is decorated by an independent copy of an auxiliary measure. These results are analogues of the results for branching Brownian motions obtained in Arguin et al. (2013), Aïdékon et al. (2013) and Roberts (2013).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:137:y:2021:i:c:p:1-34
    DOI: 10.1016/j.spa.2021.03.007
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    References listed on IDEAS

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    1. Engländer, János, 2004. "Large deviations for the growth rate of the support of supercritical super-Brownian motion," Statistics & Probability Letters, Elsevier, vol. 66(4), pages 449-456, March.
    2. Sheu, Yuan-Chung, 1997. "Lifetime and compactness of range for super-Brownian motion with a general branching mechanism," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 129-141, October.
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    Cited by:

    1. 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.
    2. Li, Zenghu & Zhu, Yaping, 2022. "Survival probability for super-Brownian motion with absorption," Statistics & Probability Letters, Elsevier, vol. 186(C).
    3. Zhang, Shuxiong, 2024. "Large deviation probabilities for the range of a d-dimensional supercritical branching random walk," Applied Mathematics and Computation, Elsevier, vol. 462(C).

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