Asymptotic moments of spatial branching processes

• Published in: Probability theory and related fields 2022-12, Vol.184 (3-4), p.805-858
• Main Authors: Gonzalez, Isaac, Horton, Emma, Kyprianou, Andreas E.
• Format: Article
• Language: English
• Subjects: Branching (mathematics); Economics; Eigenvectors; Finance; Insurance; Management; Markov analysis; Markov processes; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Polynomials; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Statistics for Business; Theoretical
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-022-01131-2

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 P δ x , 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 , lim t → ∞ g k ( t ) E δ x [ ⟨ f , X t ⟩ k ] = C k ( 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 ∫ 0 t ⟨ f , X t ⟩ d s , for bounded measurable f on E .