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 non-local branching mechanisms under a martingale change of measure. As an application of this decomposition, we obtaina 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 inKyprianou et al. (2012),Kyprianou and Murillo-Salas (2013)and Liu et al. (2009)for superprocesses with purely local branching mechanisms and inKyprianou and Palau (2018) for super Markov chains.