Asymptotic moments of spatial branching processes

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.