Asymptotic moments of spatial branching processes

Suppose that $$X =(X_t, t\ge 0)$$ 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 $${\mathbb {P}}_{\delta _x}$$ P δ x , when issued from a unit mass at $$x\in E$$ x ∈ E . For a general setting in which the first moment semigroup of X displays a Perron–Frobenius type behaviour, we show that, for $$k\ge 2$$ k ≥ 2 and any positive bounded measurable function f on E, $$\begin{aligned} \lim _{t\rightarrow \infty } g_k(t){\mathbb {E}}_{\delta _x}[\langle f, X_t\rangle ^k] = C_k(x, f), \end{aligned}$$ lim t → ∞ g k ( t ) E δ x [ ⟨ f , X t ⟩ k ] = C k ( x , f ) , where the constant $$C_k(x, f)$$ C k ( x , f ) can be identified in terms of the principal right eigenfunction and left eigenmeasure and $$g_k(t)$$ 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 $$\int _0^t \langle f, X_t \rangle \mathrm{d}s$$ ∫ 0 t ⟨ f , X t ⟩ d s , for bounded measurable f on E.