Quasi-stationary distributions for subcritical branching Markov chains

TL;DR

This paper provides a probabilistic proof for the existence of Yaglom limits in subcritical branching Markov chains, offering explicit integral formulas for all quasi-stationary distributions without using Martin boundary theory.

Contribution

It introduces a probabilistic approach using spinal decomposition and the many-to-few formula to characterize quasi-stationary distributions in subcritical branching Markov chains.

Findings

Existence of Yaglom limit proved probabilistically.

Explicit integral representations of all quasi-stationary distributions.

Proofs do not rely on Martin boundary theory.

Abstract

Consider a subcritical branching Markov chain. Let $Z_n$ denote the counting measure of particles of generation $n$. Under some conditions, we give a probabilistic proof for the existence of the Yaglom limit of $(Z_n)_{n\in\mathbb{N}}$ by the moment method, based on the spinal decomposition and the many-to-few formula. As a result, we give explicit integral representations of all quasi-stationary distributions of $(Z_n)_{n\in\mathbb{N}}$, whose proofs are direct and probabilistic, and don't rely on Martin boundary theory.