Fluctuations of non-local branching Markov processes

TL;DR

This paper investigates the fluctuation behaviors of supercritical branching Markov processes with non-local mechanisms, identifying three regimes based on spectral gap size and establishing corresponding functional central limit theorems.

Contribution

It introduces a classification of fluctuation regimes for non-local branching Markov processes and proves functional CLTs within each regime, advancing understanding of their asymptotic behavior.

Findings

Three fluctuation regimes identified based on spectral gap size

Functional central limit theorems established for each regime

Enhanced understanding of non-local branching process fluctuations

Abstract

The aim of this paper is to study the fluctuations of a general class of supercritical branching Markov processes with non-local branching mechanisms. We show the existence of three regimes according to the size of the spectral gap associated with the expectation semigroup of the branching process and establish functional central limit theorems within each regime.