Central Limit Theorems for Supercritical Branching Nonsymmetric Markov Processes

TL;DR

This paper proves a spatial central limit theorem for a broad class of supercritical, nonsymmetric Markov processes with spatially dependent branching, extending previous symmetric cases and developing new spectral theory tools.

Contribution

It generalizes existing CLTs to nonsymmetric processes and develops spectral theory for nonsymmetric semigroups, unifying previous results.

Findings

Established a spatial CLT for supercritical nonsymmetric Markov processes.

Unified previous CLTs for symmetric processes under a broader framework.

Developed spectral theory for nonsymmetric strongly continuous semigroups.

Abstract

In this paper, we establish a spatial central limit theorem for a large class of supercritical branching, not necessarily symmetric, Markov processes with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem generalizes and unifies all the central limit theorems obtained recently in \cite{RSZ2} for supercritical branching symmetric Markov processes. To prove our central limit theorem, we have to carefully develop the spectral theory of nonsymmetric strongly continuous semigroups which should be of independent interest.