Branching processes with switching

Introduction. In this report, we propose a novel modification of the Galton-Watson branching process, known as a branching process with switching.
Consider a Petri dish with a single bacterium at the initial moment under a lamp. When the lamp is on, bacterial reproduction forms a supercritical branching process. Conversely, when the light is off, the process becomes subcritical. We assume that the lamp alternates between being on and off. This procedure produces a branching process in a varying environment.
Branching processes in a varying environment are receiving significant attention nowadays. The article by Kersting G. [1] has become a key work in this field. It shows the extinction probability of these processes. The key feature of this study is the non-stationarity of the environment: the lengths of the cycles increase. This causes specific behaviour of the process. A similar model in a random environment can be found in [2].
Mathematical model. We begin with some notation. Denote by $P$ and $Q$ two discrete distributions on $\mathbb{Z}_{+}$ with $\mu^+ := {\mathbf E}_P X > 1$ and $\mu^- := {\mathbf E}_Q X < 1.$ These distributions correspond to a random variable representing the number of descendants of a single particle under the light and without it, respectively. Let $\{\tau_i\}_{i=0}^{\infty}$ be a sequence of natural numbers, and let $k$, $l$ be two natural parameters. Now, we introduce two sequences
$$ T_0^-=0,\ T_{i}^+ = T_i^- + \tau_i k,\ T_{i+1}^- = T_{i}^+ + \tau_i l, $$
where $T_i^-$ and $T_i^+$ are the $i$-th moments of the light being turned on and turned off, respectively. Define the random variables $Z_n,\;n \geq 0,$ with values in $\mathbb{N}_0$ by the relation
\begin{equation} \label{MainEq} Z_0 = 1,\ Z_{n+1} = \sum_{j=1}^{Z_n} X_{i,j}, \end{equation}
where $\{X_{i,j}\}$ are independent random variables, such that for every $i$ $\{X_{i,j}\}$ are identically distributed with the distribution $P$ for $i \in (T_l^+, T_{l}^-]$, and with the distribution $Q$ otherwise. Then the process $\{Z_n\}_{n=1}^{\infty}$ is called a branching process with switching and initial growth.
Similarly, a branching process with switching and initial decrease is defined by expression (\ref{MainEq}) with
$$ T_0^- = 0,\ T_{i}^+ = T_i^- + \tau_i l,\ T_{i+1}^- = T_{i}^+ + \tau_i k, $$
where for every $i \in (T_l^-, T_{l}^+]$ we assume that $X_{i,j}$ are distributed with the distribution $Q$ and with the distribution $P$ otherwise.
Let $\mu:=\left(\mu^+\right)^k \left(\mu^-\right)^l$. We call a branching process with switching subcritical if $\mu < 1$, critical as $\mu = 1$, and supercritical as $\mu > 1$.
The main result of this report is a theorem that describes the asymptotic behaviour of subcritical and critical branching processes with switching and its relation to Galton-Watson branching processes.

References

1. Kersting G., “A unifying approach to branching processes in a varying environment”, J. Appl. Prob., 57 (2020), 196–220
2. Korshunov I. D., “Vetvyaschiesya protsessy v sluchainoi srede s zamorazhivaniyami”, Diskretnaya matematika, 35:3 (2023), 20–36. [Korshunov, I., “Branching processes in random environment with freezing”, Discrete Math. Appl., 35:4 (2025), 235–247]