The extremal process of super-Brownian motion

Yan Xia Ren, Renming Song, Rui Zhang

Research output: Contribution to journalArticlepeer-review


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).

Original languageEnglish (US)
Pages (from-to)1-34
Number of pages34
JournalStochastic Processes and their Applications
StatePublished - Jul 2021


  • Extremal process
  • KPP equation
  • Poisson random measure
  • Super-Brownian motion
  • Supremum of the support of super-Brownian motion

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics


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