Continuous-state branching processes with collisions: First passage times and duality

We introduce a class of one-dimensional positive Markov processes generalizing continuous-state branching processes (CBs), by taking into account a phenomenon of random collisions. Besides branching, characterized by a general mechanism psi, at a constant rate in time two particles are sampled uniformly in the population, collide and leave a mass of particles governed by a (sub)critical mechanism sigma. Such CB processes with collisions (CBCs) are shown to be the only Feller processes without negative jumps satisfying a Laplace duality relationship with one-dimensional diffusions on the half-line. This generalizes the duality observed for logistic CBs in Foucart (2019). Via time-change, CBCs are also related to an auxiliary class of Markov processes, called CB processes with spectrally positive migration (CBMs), recently introduced in Vidmar (2022). We find necessary and sufficient conditions for the boundaries 0 or infinity to be attracting and for a limiting distribution to exist. The Laplace transform of the latter is provided. Under the assumption that the CBC process does not explode, the Laplace transforms of the first passage times below arbitrary levels are represented with the help of the solution of a second order differential equation, whose coefficients express in terms of the Levy-Khintchine functions sigma and psi. Sufficient conditions for non-explosion are given.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

This study introduces a class of one-dimensional positive Markov processes that generalize continuous-state branching processes by incorporating random collisions. The study establishes that these processes, known as CB processes with collisions (CBCs), are the only Feller processes without negative jumps that satisfy a Laplace duality relationship with one-dimensional diffusions. The study also explores the relationship between CBCs and CB processes with spectrally positive migration, and provides necessary and sufficient conditions for attracting boundaries and the existence of a limiting distribution.