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Abstract
Suppose that X= (X _{t}, t≥ 0) is either a superprocess or a branching Markov process on a general space E, with nonlocal 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 

Pages (fromto)  805858 
Number of pages  54 
Journal  Probability Theory and Related Fields 
Volume  184 
Issue number  34 
Early online date  25 Apr 2022 
DOIs  
Publication status  Published  31 Dec 2022 
Keywords
 Asymptotic behaviour
 Branching processes
 Moments
 Nonlocal branching
 Superprocesses
ASJC Scopus subject areas
 Analysis
 Statistics and Probability
 Statistics, Probability and Uncertainty
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Mathematical Theory of Radiation Transport: Nuclear Technology Frontiers (MATHRAD):
Kyprianou, A., Cox, A., Pryer, T. & Hattam, L.
Engineering and Physical Sciences Research Council
1/09/22 → 30/11/27
Project: Research council

Stochastic Analysis of the Neutron Transport Equation and Applications to Nuclear Safety
Kyprianou, A., Cox, A. & Harris, S.
Engineering and Physical Sciences Research Council
16/05/17 → 31/12/21
Project: Research council