Yaglom limit for critical nonlocal branching Markov processes

Abstract

We consider the classical Yaglom limit theorem for a branching Markov process $\mathit{X}=({\mathit{X}}_{\mathit{t}},\mathit{t}\ge 0)$, with nonlocal branching mechanism in the setting that the mean semigroup is critical, that is, its leading eigenvalue is zero. In particular, we show that there exists a constant $\mathit{c}(\mathit{f})$ such that

$$\mathrm{Law}(\frac{⟨\mathit{f},{\mathit{X}}_{\mathit{t}}⟩}{\mathit{t}}|⟨1,{\mathit{X}}_{\mathit{t}}⟩>0)\to {\mathbf{e}}_{\mathit{c}(\mathit{f})},\mathit{t}\to \infty ,$$

where ${\mathbf{e}}_{\mathit{c}(\mathit{f})}$ is an exponential random variable with rate $\mathit{c}(\mathit{f})$ 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, (Probab. Theory Related Fields 173 (2019) 999–1062; Electron. Commun. Probab. 23 (2018) 42), these results do not allow for nonlocal 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 kth martingale moments of X (rather than the Yaglom limit itself). We then illustrate our results in the setting of neutron transport for which the nonlocality is essential, complementing recent developments in this domain (Ann. Appl. Probab. 30 (2020) 2573–2612; Ann. Appl. Probab. 30 (2020) 2815–2845; SIAM J. Appl. Math. 81 (2021) 982–1001; Cox et al. (2021); J. Stat. Phys. 176 (2019) 425–455).