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

TL;DR

This paper introduces a new class of Markov processes called CBCs that incorporate random collisions into continuous-state branching processes, establishing duality relationships, boundary behaviors, and explicit formulas for first passage times.

Contribution

It generalizes continuous-state branching processes by including collisions, characterizes their duality with diffusions, and provides conditions for boundary behaviors and first passage time distributions.

Findings

CBCs are the only Feller processes without negative jumps satisfying the duality.

Necessary and sufficient conditions for boundary attraction and existence of limiting distributions are established.

Laplace transforms of first passage times are expressed via solutions to differential equations.

Abstract

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 by Foucart. Via time-change, CBCs are also related to an auxiliary class of Markov processes, called CB processes with spectrally positive migration (CBMs), recently introduced by Vidmar. We find necessary and sufficient conditions for the boundaries $0$ or $\infty$ 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 L\'evy-Khintchine functions $\Sigma$ and $\Psi$. Sufficient conditions for non-explosion are given.