Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fluctuations of non-local branching Markov processes

Christopher B. C. Dean, Emma Horton

TL;DR

The paper analyzes fluctuations of a broad class of supercritical branching Markov processes with non-local branching on a Polish space, establishing a functional central limit theorem with a trichotomy determined by the spectral gap of the first-moment semigroup. It develops a unified spectral framework that allows non-simple leading eigenvalues and derives three regimes: large, small, and critical, each with distinct limiting behavior. In the large regime, fluctuations around the leading eigenstructure vanish after appropriate scaling, while in the small and critical regimes the centered fluctuations converge to Gaussian processes with explicitly defined covariances. The results extend previous work to non-local branching and non-simple leading eigenvalues, providing precise descriptions of long-time fluctuations with potential oscillations in the critical case.

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.

Fluctuations of non-local branching Markov processes

TL;DR

The paper analyzes fluctuations of a broad class of supercritical branching Markov processes with non-local branching on a Polish space, establishing a functional central limit theorem with a trichotomy determined by the spectral gap of the first-moment semigroup. It develops a unified spectral framework that allows non-simple leading eigenvalues and derives three regimes: large, small, and critical, each with distinct limiting behavior. In the large regime, fluctuations around the leading eigenstructure vanish after appropriate scaling, while in the small and critical regimes the centered fluctuations converge to Gaussian processes with explicitly defined covariances. The results extend previous work to non-local branching and non-simple leading eigenvalues, providing precise descriptions of long-time fluctuations with potential oscillations in the critical case.

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.

Paper Structure

This paper contains 8 sections, 25 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proofs of the main results
  4. Large Regime
  5. Small regime
  6. Critical regime
  7. Appendix
  8. Appendix

Key Result

Theorem 1

Suppose that $2\lambda_2>\lambda_1$ and let $f \in B(E)$. Then, for any initial position $x \in E$, as $t\rightarrow \infty$ $\mathbb{P}_{\delta_x}$-almost surely and in $L^2$, where $\widetilde{W}_{\infty}(x)$ is the $\mathbb{P}_{\delta_x}$-almost sure limit of the martingale

Theorems & Definitions (45)

  • Theorem : Large Regime (simplified)
  • Theorem : Small Regime (simplified)
  • Theorem : Critical Regime (simplified)
  • Theorem 2.1: Large Regime
  • Remark 2.2
  • Corollary 2.3: SLLN
  • Remark 2.4
  • Theorem 2.5: Small Regime
  • Remark 2.6
  • Theorem 2.7: Critical Regime
  • ...and 35 more