TY - JOUR
T1 - The extremal process of super-Brownian motion
AU - Ren, Yan Xia
AU - Song, Renming
AU - Zhang, Rui
N1 - Funding Information:
We thank the referee for helpful comments. Part of the research for this paper was done while the second-named author was visiting Jiangsu Normal University, where he was partially supported by a grant from the National Natural Science Foundation of China ( 11931004 ) and by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
Funding Information:
The research of this author is supported by NSFC, PR China (Grant No. 11671017, 11731009 and 12071011) and LMEQF, PR China.Research supported in part by a grant from the Simons Foundation, USA (#429343, Renming Song).The research of this author is supported by NSFC, PR China (Grant No. 11601354), Beijing Municipal Natural Science Foundation, PR China (Grant No. 1202004), and Academy for Multidisciplinary Studies, Capital Normal University, PR China.We thank the referee for helpful comments. Part of the research for this paper was done while the second-named author was visiting Jiangsu Normal University, where he was partially supported by a grant from the National Natural Science Foundation of China (11931004) and by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
Publisher Copyright:
© 2021 Elsevier B.V.
PY - 2021/7
Y1 - 2021/7
N2 - 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).
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).
KW - Extremal process
KW - KPP equation
KW - Poisson random measure
KW - Super-Brownian motion
KW - Supremum of the support of super-Brownian motion
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U2 - 10.1016/j.spa.2021.03.007
DO - 10.1016/j.spa.2021.03.007
M3 - Article
AN - SCOPUS:85103971301
SN - 0304-4149
VL - 137
SP - 1
EP - 34
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
ER -