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.