Stationary measures and the continuous-state branching process conditioned on extinction

TL;DR

This paper studies continuous-state branching processes that almost surely become extinct, characterizing their stationary measures, and analyzing their conditioned limit distributions, including size-biased and Yaglom distributions.

Contribution

It provides a representation of stationary measures via Lévy process scale functions and establishes limit theorems for conditioned processes, connecting to size-biased and Yaglom distributions.

Findings

Stationary measures are represented through Lévy process scale functions.

Limit theorems describe the distribution of processes conditioned on extinction.

Critical and subcritical cases yield different non-degenerate limit distributions.

Abstract

We consider continuous-state branching processes (CB processes) which become extinct almost surely. First, we tackle the problem of describing the stationary measures on $(0,+\infty)$ for such CB processes. We give a representation of the stationary measure in terms of scale functions of related L\'{e}vy processes. Then we prove that the stationary measure can be obtained from the vague limit of the potential measure, and, in the critical case, can also be obtained from the vague limit of a normalized transition probability. Next, we prove some limit theorems for the CB process conditioned on extinction in a near future and on extinction at a fixed time. We obtain non-degenerate limit distributions which are of the size-biased type of the stationary measure in the critical case and of the Yaglom's distribution in the subcritical case. Finally we explore some further properties of the limit distributions.