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 languageEnglish
Pages (from-to)805-858
Number of pages54
JournalProbability Theory and Related Fields
Volume184
Issue number3-4
Early online date25 Apr 2022
DOIs
Publication statusPublished - 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

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