## Abstract

Suppose that X= (X
_{t}, t≥ 0) is either a superprocess or a branching Markov process on a general space E, with non-local branching mechanism and probabilities Pδx, when issued from a unit mass at x∈ E. For a general setting in which the first moment semigroup of X displays a Perron–Frobenius type behaviour, we show that, for k≥ 2 and any positive bounded measurable function f on E, limt→∞gk(t)Eδx[⟨f,Xt⟩k]=Ck(x,f),where the constant C
_{k}(x, f) can be identified in terms of the principal right eigenfunction and left eigenmeasure and g
_{k}(t) is an appropriate deterministic normalisation, which can be identified explicitly as either polynomial in t or exponential in t, depending on whether X is a critical, supercritical or subcritical process. The method we employ is extremely robust and we are able to extract similarly precise results that additionally give us the moment growth with time of ∫0t⟨f,Xt⟩ds, for bounded measurable f on E.

Original language | English |
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Pages (from-to) | 805-858 |

Number of pages | 54 |

Journal | Probability Theory and Related Fields |

Volume | 184 |

Issue number | 3-4 |

Early online date | 25 Apr 2022 |

DOIs | |

Publication status | Published - 31 Dec 2022 |

## Keywords

- Asymptotic behaviour
- Branching processes
- Moments
- Non-local branching
- Superprocesses

## ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty