Yaglom limit for critical neutron transport

Abstract

We consider the classical Yaglom limit theorem for a branching Markov process $X = (X_t, t \ge 0)$, with non-local branching mechanism in the setting that the mean semigroup is critical, i.e. its leading eigenvalue is zero. In particular, we show that there exists a constant $c(f)$ such that \[ {\rm Law}\left(\frac{\langle f, X_t\rangle}{t} \bigg| \langle 1, X_t\rangle > 0 \right) \to {\mathbf e}_{c(f)}, \qquad t \to \infty, \] where ${\mathbf e}_{c(f)}$ is an exponential random variable with rate $c(f)$ and the convergence is in distribution. As part of the proof, we also show that the probability of survival decays inversely proportionally to time. Although Yaglom limit theorems have recently been handled in the setting of branching Brownian motion in a bounded domain and superprocesses, \cite{Ellen, Yanxia}, these results do not allow for non-local branching, which complicates the analysis. Our approach and the main novelty of this work is based around a precise result for the scaled asymptotics for the $k$-th martingale moments of $X$ (rather than the Yaglom limit itself). We then illustrate our results in the setting of neutron transport, for which the non-locality is essential, complementing recent developments in this domain \cite{SNTE, SNTEII, SNTEIII, MCNTE, MultiNTE}.