More on the asymptotic behaviour of moments of branching Markov processes

TL;DR

This paper develops a comprehensive asymptotic analysis of the higher-order moments of a measure-valued branching Markov process with non-local branching, allowing a non-simple leading eigenvalue and multiple spectral components. It constructs a generalized spectral framework with eigenvalues $\lambda_i$, a nilpotent operator $\mathcal{N}$, and associated eigenfunctions to obtain precise moment asymptotics across large, critical, and small growth regimes, including explicit leading terms $L_A$ and uniform error bounds. The main contributions are the rigorous, regime-by-regime descriptions of $\psi_t^{(k)}[\boldsymbol{f}]$ for $\boldsymbol{f}$ in spectral subspaces $\mathrm{Ei}(\Lambda_L)$, $\mathrm{Ei}(\Lambda_C)$, and $\mathrm{Ei}(\Lambda_S)$, plus inductive proof techniques leveraging a branching-structure operator $\zeta_{[k]}$ and an evolution equation relating higher-order moments to lower-order ones. These results lay the groundwork for central limit theorems and extended LLNs by providing the necessary fluctuation structure and moment control for $X_t$.

Abstract

Consider a branching Markov process, $X = (X(t), t \ge 0)$, with non-local branching mechanism. Studying the asymptotic behaviour of the moments of X has recently received attention in the literature [6, 7] due to the importance of these results in understanding the underlying genealogical structure of $X$. In this article, we generalise the results of [7] to allow for a non-simple leading eigenvalue and to also study the higher order fluctuations of the moments of $X$. These results will be useful for proving central limit theorems and extending well-known LLN results.

Theorems & Definitions (11)

• Theorem 2.1: Large Regime
• Theorem 2.2: Small Regime
• Remark 2.3
• Theorem 2.4: Critical Regime
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 4.1