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Quasi-stationary distributions for subcritical population models

Pablo Groisman, Leonardo T. Rolla, Célio Terra

Abstract

Subcritical population processes are attracted to extinction and do not have 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 characteristics of the process that may prevent it from possessing multiple QSDs are not entirely clear. 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 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.

Quasi-stationary distributions for subcritical population models

Abstract

Subcritical population processes are attracted to extinction and do not have 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 characteristics of the process that may prevent it from possessing multiple QSDs are not entirely clear. 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 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.

Paper Structure

This paper contains 6 sections, 7 theorems, 31 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The branching process with genealogy
  3. The QSD for the branching process with genealogy
  4. Uniqueness of QSD for branching random walks
  5. $\alpha$-positiveness and the Yaglom limit
  6. The Q-process

Key Result

Proposition 1.1

Let $(\xi_t)_{t \ge 0}$ be an continuous-time Markov chain on a countable state-space $\Lambda_0:=\Lambda \cup \{\emptyset\}$ absorbed at $\emptyset$ almost surely, such that $(\xi_t)_{t \ge 0}$ restricted to $\Lambda$ is irreducible. Suppose that there is a projection $\pi: \Lambda_0 \rightarrow \m

Figures (3)

  • Figure 1: Relationships between measures
  • Figure 2: Two examples of pruning. The dashed piece of the tree is deleted when vertex $x$ dies.
  • Figure 3: If there are two particles at time $0$ with alive descendants at time $t$, there are particles very far from each other at time $t$

Theorems & Definitions (14)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • proof : Proof of Theorem \ref{['thm_unique QSD']}
  • proof : Proof of Theorem \ref{['thm_uniqueQSD_BRW']}
  • Lemma 4.1
  • proof : Proof of Lemma \ref{['lemma_jumpwalker']}.
  • Theorem 5.1: Kingman63
  • proof : Proof of Proposition \ref{['prop_Yaglom']}.
  • ...and 4 more