The extremal process of super-Brownian motion

Yan Xia Ren; Renming Song; Rui Zhang

doi:10.1016/j.spa.2021.03.007

The extremal process of super-Brownian motion

Yan Xia Ren, Renming Song, Rui Zhang

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we establish limit theorems for the supremum of the support, denoted by M_t, of a supercritical super-Brownian motion {X_t,t≥0} on R. We prove that there exists an m(t) such that (X_t−m(t),M_t−m(t)) converges in law, and give some large deviation results for M_t as t→∞. We also prove that the limit of the extremal process E_t≔X_t−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 language	English (US)
Pages (from-to)	1-34
Number of pages	34
Journal	Stochastic Processes and their Applications
Volume	137
DOIs	https://doi.org/10.1016/j.spa.2021.03.007
State	Published - Jul 2021

Keywords

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

Online availability

10.1016/j.spa.2021.03.007

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title = "The extremal process of super-Brownian motion",

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AU - Zhang, Rui

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AB - 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).

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The extremal process of super-Brownian motion

Abstract

Keywords

ASJC Scopus subject areas

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