On branching process with a threshold and its generalization

Jishi Sun The Affiliated High School of Peking University

S.-T. Yau High School Science Awarded Papers mathscidoc:1801.35008

Yau Science Award (Math), 2017.12
This paper studies a special class of population-size-dependent branching processes, in which the offspring distribution is supercritical when the population size does not exceed a given threshold ๐พ๐พ, and is subcritical or critical when the population size exceeds ๐พ๐พ. Up to now, the author has found no paper concerning continuous time threshold processes, and study of the discrete case is also limited to the extinction time ๐‘‡๐‘‡ and ๐”ผ๐”ผ๐‘‡๐‘‡. In this paper, both of the two cases were studied more thoroughly: for the discrete case, a limit theorem of the process has been proved; for the continuous case, the existence of Markovian threshold processes has been proved, and an equivalent condition for the process to extinct almost surely was also given. This paper also further generalized discrete time threshold processes by giving a random delay to the thresholdโ€™s influence of offspring distribution. The properties of these delayed threshold processes were studied, and a limit theorem was obtained.
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  title={On branching process with a threshold and its generalization},
  author={Jishi Sun},
  booktitle={Yau Science Award (Math)},
Jishi Sun. On branching process with a threshold and its generalization. 2017. In Yau Science Award (Math). http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180113194956004178881.
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