Many-to-few for non-local branching Markov process

Harris, Simon C.; Horton, Emma; Kyprianou, Andreas E.; Powell, Ellen

Abstract

We provide a many-to-few formula in the general setting of non-local branching Markov processes. This formula allows one to compute expectations of k-fold sums over functions of the population at k different times. The result generalises [13] to the non-local setting, as introduced in [11] and [8]. As an application, we consider the case when the branching process is critical, and conditioned to survive for a large time. In this setting, we prove a general formula for the limiting law of the death time of the most recent common ancestor of two particles selected uniformly from the population at two different times, as t→∞. Moreover, we describe the limiting law of the population sizes at two different times, in the same asymptotic regime.

Citation

Harris, S. C., Horton, E., Kyprianou, A. E., & Powell, E. (2024). Many-to-few for non-local branching Markov process. Electronic Journal of Probability, 29, 1-26. https://doi.org/10.1214/24-ejp1098

• Journal Article Type: Article
• Acceptance Date: Feb 9, 2024
• Online Publication Date: Mar 4, 2024
• Publication Date: 2024
• Deposit Date: May 28, 2024
• Publicly Available Date: May 28, 2024
• Journal: Electronic Journal of Probability
• Electronic ISSN: 1083-6489
• Publisher: Institute of Mathematical Statistics
• Peer Reviewed: Peer Reviewed
• Volume: 29
• Article Number: 41
• Pages: 1-26
• DOI: https://doi.org/10.1214/24-ejp1098
• Public URL: https://durham-repository.worktribe.com/output/2466682