$L\log L$ criterion for a class of multitype superdiffusions with non-local branching mechanisms

TL;DR

This paper establishes an $L\,\log\,L$ criterion for multitype superdiffusions with non-local branching, providing a pathwise spine decomposition and a condition for the martingale limit to be non-degenerate.

Contribution

It introduces a novel $L\log L$ criterion and a pathwise spine decomposition for multitype superdiffusions with non-local branching mechanisms.

Findings

Derived a necessary and sufficient $L\log L$ condition for martingale non-degeneracy.

Extended superprocess results to multitype superdiffusions with non-local branching.

Provided a framework applicable to super Markov chains and related processes.

Abstract

In this paper, we provide a pathwise spine decomposition for multitype superdiffusions with non-local branching mechanisms under a martingale change of measure. As an application of this decomposition, we obtain a necessary and sufficient condition (called the $L\log L$ criterion) for the limit of the fundamental martingale to be non-degenerate. This result complements the related results for superprocesses with purely local branching mechanisms and in Kyprianou and Palau (2016) for super Markov chains.