IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v137y2021icp1-34.html
   My bibliography  Save this article

The extremal process of super-Brownian motion

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414921000351
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2021.03.007?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Li, Zenghu & Zhu, Yaping, 2022. "Survival probability for super-Brownian motion with absorption," Statistics & Probability Letters, Elsevier, vol. 186(C).
    2. 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).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kyprianou, A.E. & Ren, Y.-X., 2012. "Backbone decomposition for continuous-state branching processes with immigration," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 139-144.
    2. Berestycki, J. & Kyprianou, A.E. & Murillo-Salas, A., 2011. "The prolific backbone for supercritical superprocesses," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1315-1331, June.
    3. Li, Zenghu & Zhu, Yaping, 2022. "Survival probability for super-Brownian motion with absorption," Statistics & Probability Letters, Elsevier, vol. 186(C).
    4. Engländer, János & Fleischmann, Klaus, 2000. "Extinction properties of super-Brownian motions with additional spatially dependent mass production," Stochastic Processes and their Applications, Elsevier, vol. 88(1), pages 37-58, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:137:y:2021:i:c:p:1-34. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.