On the uniqueness of quasi-stationary distributions for population models with spatial structure

Subcritical population processes are attracted to extinction and do not have non-trivial stationary distributions, which prompts the study of quasi-stationary distributions (QSDs) instead. In contrast to what generally happens for stationary distributions, QSDs may not be unique, even under irreducibility conditions. The general conditions for uniqueness of QSDs are not always easy to check. For the branching process, besides the quasi-limiting distribution there are many other QSDs. In this paper, we investigate whether adding little extra information to the continuous-time branching process is enough to obtain uniqueness. We consider the branching process with genealogy and branching random walks, and show that they have a unique QSD.

💡 Research Summary

The paper investigates the uniqueness of quasi‑stationary distributions (QSDs) for subcritical population models when additional genealogical or spatial information is incorporated. In a subcritical continuous‑time branching process, extinction occurs almost surely and the process admits a whole family of QSDs, distinguished by their mean absorption times and tail heaviness. Classical criteria for QSD uniqueness—such as finiteness of the state space, rapid attraction to a finite set, or exponential integrability of the absorption time—do not apply to the plain branching process, and indeed the process can have infinitely many QSDs.

The authors ask whether a modest enrichment of the model’s structure can enforce uniqueness. They study two extensions:

1. Branching Process with Genealogy (BPG).
   Each alive individual is represented by a leaf of a rooted oriented tree. When a leaf’s exponential clock rings, an independent copy of the offspring variable (Z) is drawn; (Z) new children are attached to the leaf. After each birth event a pruning operation removes vertices that are no longer needed to determine the ancestry of the current leaves. The pruning rules guarantee that the root is always the most recent common ancestor of all leaves. The projection of the tree onto the number of leaves reproduces the ordinary branching process.

2. Branching Random Walk modulo translations (BRW/∼).
   Particles live on (\mathbb{Z}^d). Each particle dies at rate 1 and gives birth at rate (\lambda<1) to a child placed uniformly at a neighboring site. The state space (\Delta^d) consists of finite particle configurations. By identifying configurations that differ only by a global translation, the authors obtain the quotient space (\Delta^d_{\sim}). The induced process is translation‑invariant and its projection onto the total particle count is a subcritical branching process with offspring distribution supported on ({0,2}).

Both models are built on a light‑tailed subcritical offspring distribution: (\mathbb{E}