The survival probability of critical and subcritical branching processes in finite state space Markovian environment

TL;DR

This paper extends classical branching process classifications to Markovian environments, analyzing the asymptotic survival probabilities in finite state spaces, revealing new behaviors in critical and subcritical regimes.

Contribution

It generalizes the classification of branching processes from i.i.d. to Markovian environments and studies their long-term survival probabilities.

Findings

Asymptotic survival probabilities are characterized for different regimes.

New classifications for critical and subcritical states in Markovian environments.

Results extend classical theory to more complex, dependent environments.

Abstract

Let $(Z_n)_{n\geqslant 0}$ be a branching process in a random environment defined by a Markov chain $(X_n)_{n\geqslant 0}$ with values in a finite state space $\mathbb X$ starting at $X_0=i \in\mathbb X$. We extend from the i.i.d. environment to the Markovian one the classical classification of the branching processes into critical and strongly, intermediate and weakly subcritical states. In all these cases, we study the asymptotic behaviour of the probability that $Z_n>0$ as $n\to+\infty$.