Stability of (sub)critical nonlocal spatial branching processes with and without immigration

We consider the setting of either a general nonlocal branching particle process or a general nonlocal superprocess, in both cases, with and without immigration. Under the assumption that the mean semigroup has a Perron–Frobenious-type behaviour, as well as the existence of second moments, we consider necessary and sufficient conditions that ensure limiting distributional stability. More precisely, our first main contribution pertains to proving the asymptotic Kolmogorov survival probability and Yaglom limit for critical nonlocal branching particle systems and super-processes under a second moment assumption on the offspring distribution. Our results improve on existing literature by removing the requirement of bounded offspring in the particle setting (Ann. Probab. 50 (2022) 2373–2408) and generalising (Acta Appl. Math. 165 (2020) 91–131) to allow for nonlocal branching mechanisms. Our second main contribution pertains to the stability of both critical and sub-critical nonlocal branching particle systems and super-processes with immigration. At criticality, we show that the scaled process converges to a Gamma distribution under a necessary and sufficient integral test. At subcriticality we show stability of the process, also subject to an integral test. In these cases, our results complement classical results for (continuous-time) Galton–Watson processes with immigration and continuous-state branching processes with immigration; see (J. Roy. Statist. Soc. Ser. B, Methodol. 27 (1965) 138–143; Bull. Amer. Math. Soc. 78 (1972) 242–244; J. Appl. Probab. 7 (1970) 411–422; Teor. Veroyatn. Primen. 2 (1957) 339–348; J. Appl. Probab. 9 (1972) 24–31), among others. In the setting of super-processes, the only work we know of at this level of generality is summarised in (Measure-Valued Branching Markov Processes (2022) Springer). The proofs of our results, both with and without immigration, appeal to similar technical approaches and accordingly, we include the results together in this paper.