In this talk, we consider the setting of either a general non-local branching particle process or a general non-local superprocess. Under the assumption that the mean semigroup has a Perron-Frobenious type behaviour in combination with a regularly varying assumption on the reproductive point process, which permits infinite second moments, we consider sufficient conditions that ensure limiting distributional stability when conditioned on survival at criticality. We offer two main results. Under the aforesaid conditions, our first main contribution establishes the polynomial decay in time of the survival probability in the spirit of a classical Kolmogorov limit. The second main contribution pertains to the stability, when conditioning on survival, in the spirit of a Yaglom limit. This is joint work with Andreas E. Kyprianou and Pedro Mart´ın-Ch´avez (University of Warwick).
Abstract. abs