Abstract
A continuous-state branching process in varying environments is constructed by the pathwise unique positive solution to a stochastic integral equation driven by time-space noises. The cumulant semigroup of the process is characterized in terms of a backward integral equation. We clarify the behavior of the process at its bottlenecks, which are the deterministic times when it arrives at zero almost surely by negative jumps. The process arises naturally as the scaling limit of Galton–Watson processes in varying environments.
Funding Statement
The research was supported by the National Key R&D Program of China (No. 2020YFA0712900), the National Natural Science Foundation of China (No. 11531001), the Program for Probability and Statistics: Theory and Application (No. IRTL1704) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ).
Acknowledgments
The authors would like to thank Peisen Li and two referees for very helpful comments on earlier versions of the work.
Citation
Rongjuan Fang. Zenghu Li. "Construction of continuous-state branching processes in varying environments." Ann. Appl. Probab. 32 (5) 3645 - 3673, October 2022. https://doi.org/10.1214/21-AAP1770
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