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

In this paper, we provide a pathwise spine decomposition for multitype superdiffusions with nonlocal 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 obtained in Kyprianou et al. (2012), Kyprianou and Murillo-Salas (2013) and Liu et al. (2009) for superprocesses with purely local branching mechanisms and in Kyprianou and Palau (2018) for super Markov chains.