Asymptotic moments of spatial branching processes

Abstract

Suppose that X =(X_t, tge 0) is either a superprocess or a branching Markov process on a general space E, with non-local branching mechanism and probabilities {mathbb {P}}_{delta _x}, when issued from a unit mass at xin E. For a general setting in which the first moment semigroup of X displays a Perron–Frobenius type behaviour, we show that, for kge 2 and any positive bounded measurable function f on E, limt→∞gk(t)Eδx[⟨f,Xt⟩k]=Ck(x,f),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\lim _{t\\rightarrow \\infty } g_k(t){\\mathbb {E}}_{\\delta _x}[\\langle f, X_t\\rangle ^k] = C_k(x, f), \\end{aligned}$$\\end{document}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 int _0^t langle f, X_t rangle mathrm{d}s, for bounded measurable f on E.