[Paper Review] The critical Branching Markov Chain is transient

Nina Gantert, Sebastian Mueller|arXiv (Cornell University)|Oct 26, 2005

Stochastic processes and statistical mechanics6 references31 citations

TL;DR

This paper establishes that the critical Branching Markov Chain (BMC) is transient, meaning no state is visited infinitely often almost surely. It proves transience using spectral radius analysis and superharmonic functions, showing that when the mean offspring number equals the reciprocal of the spectral radius, the process still fails to return to any state infinitely, even under irreducible motion and constant branching rates.

ABSTRACT

We investigate recurrence and transience of Branching Markov Chains (BMC) in discrete time. Branching Markov Chains are clouds of particles which move (according to an irreducible underlying Markov Chain) and produce offspring independently. The offspring distribution can depend on the location of the particle. If the offspring distribution is constant for all locations, these are Tree-Indexed Markov chains in the sense of \cite{benjamini94}. Starting with one particle at location $x$, we denote by $α(x)$ the probability that $x$ is visited infinitely often by the cloud. Due to the irreducibility of the underlying Markov Chain, there are three regimes: either $α(x) = 0$ for all $x$ (transient regime), or $0 < α(x) < 1$ for all $x$ (weakly recurrent regime) or $α(x) = 1$ for all $x$ (strongly recurrent regime). We give classification results, including a sufficient condition for transience in the general case. If the mean of the offspring distribution is constant, we give a criterion for transience involving the spectral radius of the underlying Markov Chain and the mean of the offspring distribution.

Motivation & Objective

• To classify the recurrence and transience behavior of Branching Markov Chains (BMCs) with general, location-dependent offspring distributions.
• To determine whether the critical BMC—where the mean offspring number equals the reciprocal of the spectral radius—exhibits recurrence or transience.
• To establish sufficient conditions for transience in general BMCs using superharmonic functions and spectral radius theory.
• To show that under quasi-transitive or uniform convergence assumptions, the weakly recurrent regime does not occur, and recurrence implies strong recurrence.

Proposed method

• Define the spectral radius $\rho(P)$ of the underlying irreducible Markov chain $P$ as $\limsup_{n\to\infty} p^{(n)}(x,y)^{1/n}$, which characterizes the decay rate of return probabilities.
• Use $t$-superharmonic functions $f$ satisfying $Pf \leq t f$ to characterize $\rho(P)$ as the infimum of $t > 0$ for which such positive functions exist.
• Construct a sequence of embedded Galton-Watson processes by observing the BMC at regular intervals $k_i$ where return probabilities $p^{(k_i)}(x_{s_i},x_{s_i})$ exceed $m^{-k_i}$, ensuring supercriticality.
• Show that extinction probabilities of these embedded processes are uniformly bounded away from 1 by exploiting finitely many orbits under quasi-transitivity, implying positive probability of infinite returns.
• Apply the Perron-Frobenius theorem to compute $\rho(P)$ for symmetric random walks on $\mathbb{Z}^d$, yielding $\rho(P) = 2\sum_{i=1}^d \sqrt{p_i^+ p_i^-}$.
• Derive a critical threshold $m_c = 1 / \rho(P)$ such that the BMC is transient if $m \leq m_c$ and strongly recurrent if $m > m_c$.

Experimental results

Research questions

• RQ1 Is the critical Branching Markov Chain, where the mean offspring number equals $1/\rho(P)$, transient or recurrent?
• RQ2 Under what general conditions on the offspring distribution and underlying Markov chain can transience be guaranteed?
• RQ3 Does the weakly recurrent regime—where $0 < \alpha(x) < 1$ for all $x$—actually occur, or is it ruled out under symmetry or uniformity assumptions?
• RQ4 Can the recurrence/transience classification be reduced to a spectral radius condition when the offspring mean is constant?
• RQ5 How does the structure of the underlying Markov chain (e.g., symmetric random walk on $\mathbb{Z}^d$) affect the critical branching intensity?

Key findings

• The critical Branching Markov Chain is transient: when the mean offspring number $m = 1/\rho(P)$, the probability $\alpha(x)$ that any state $x$ is visited infinitely often is zero for all $x$.
• For a symmetric random walk on $\mathbb{Z}^d$, the BMC is strongly recurrent if $m > 1 / \left(2 \sum_{i=1}^d \sqrt{p_i^+ p_i^-} \right)$, and transient otherwise.
• In the case of a random walk on $\mathbb{Z}$ with drift $p \in (0,1)$, the BMC is transient if $m \leq 1 / (2\sqrt{p(1-p)})$, and strongly recurrent if $m > 1 / (2\sqrt{p(1-p)})$.
• The weakly recurrent regime does not occur under quasi-transitivity: if the underlying chain is quasi-transitive and the offspring mean is constant, then either $\alpha(x) = 0$ for all $x$ (transient) or $\alpha(x) = 1$ for all $x$ (strongly recurrent).
• The extinction probabilities of embedded Galton-Watson processes constructed at return times are uniformly bounded away from 1 when the spectral radius condition is met, ensuring positive probability of infinite returns if transience were to fail.
• The result resolves a discrepancy in prior work: the inequality in Theorem 4.3 of [3] should be $\leq$ rather than $<$, confirming that transience holds at the critical threshold.