Laws of Large Numbers for Supercritical Branching Gaussian Processes

TL;DR

This paper introduces a broad class of supercritical Gaussian branching particle systems, analyzing their long-term behavior through laws of large numbers, with applications to processes exhibiting long memory and roughness.

Contribution

It develops both weak and strong laws of large numbers for non-Markov Gaussian branching systems, characterizing limits via particle motion and mutation.

Findings

Established laws of large numbers for supercritical Gaussian branching processes.

Included long memory and rough processes as key examples.

Utilized moment measure techniques for analysis.

Abstract

A general class of non-Markov, supercritical Gaussian branching particle systems is introduced and its long-time asymptotics is studied. Both weak and strong laws of large numbers are developed with the limit object being characterized in terms of particle motion/mutation. Long memory processes, like branching fractional Brownian motion and fractional Ornstein-Uhlenbeck processes with large Hurst parameters, as well as rough processes, like fractional processes with with smaller Hurst parameter, are included as important examples. General branching with second moments is allowed and moment measure techniques are utilized.