TY - JOUR
T1 - Lower Deviation for the Supremum of the Support of Super-Brownian Motion
AU - Ren, Yan Xia
AU - Song, Renming
AU - Zhang, Rui
N1 - The research of this project is supported by the National Key R &D Program of China (No. 2020YFA0712900). Yan-Xia Ren is supported by NSFC (Grant Nos. 11731009, 12071011 and 12231002) and the Fundamental Research Funds for Central Universities, Peking University LMEQF. Renming Song\u2019s research was supported in part by a grant from the Simons Foundation (#960480, Renming Song). Rui Zhang is supported by NSFC (Grant Nos. 11601354 and 12271374), Beijing Municipal Natural Science Foundation (Grant No. 1202004), and Academy for Multidisciplinary Studies, Capital Normal University. We thank the referee for very 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, Yingchao Xie).
PY - 2024/6
Y1 - 2024/6
N2 - We study the asymptotic behavior of the supremum Mt of the support of a supercritical super-Brownian motion. In our recent paper (Ren et al. in Stoch Proc Appl 137:1–34, 2021), we showed that, under some conditions, Mt-m(t) converges in distribution to a randomly shifted Gumbel random variable, where m(t)=c0t-c1logt. In the same paper, we also studied the upper large deviation of Mt, i.e., the asymptotic behavior of P(Mt>δc0t) for δ≥1. In this paper, we study the lower large deviation of Mt, i.e., the asymptotic behavior of P(Mt≤δc0t|S) for δ<1, where S is the survival event.
AB - We study the asymptotic behavior of the supremum Mt of the support of a supercritical super-Brownian motion. In our recent paper (Ren et al. in Stoch Proc Appl 137:1–34, 2021), we showed that, under some conditions, Mt-m(t) converges in distribution to a randomly shifted Gumbel random variable, where m(t)=c0t-c1logt. In the same paper, we also studied the upper large deviation of Mt, i.e., the asymptotic behavior of P(Mt>δc0t) for δ≥1. In this paper, we study the lower large deviation of Mt, i.e., the asymptotic behavior of P(Mt≤δc0t|S) for δ<1, where S is the survival event.
KW - Lower large deviation
KW - Super-Brownian motion
KW - Supremum of support
UR - http://www.scopus.com/inward/record.url?scp=85174484380&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85174484380&partnerID=8YFLogxK
U2 - 10.1007/s10959-023-01292-3
DO - 10.1007/s10959-023-01292-3
M3 - Article
AN - SCOPUS:85174484380
SN - 0894-9840
VL - 37
SP - 1079
EP - 1123
JO - Journal of Theoretical Probability
JF - Journal of Theoretical Probability
IS - 2
ER -