Asymptotic moments of spatial branching processes

Abstract

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.

Many-to-few for non-local branching Markov process

“…genealogical coalescent structure, [12], martingale convergence, [10], maximal displacement of extreme particles, [17], the structure of level sets for branching Brownian motion, [4] and the analysis of certain models from the theory of stochastic genetics, [7]. We further remark that the many-to-few formula is related to recent work pertaining to asymptotic moment convergence in [8].…”

“…This formula allows one to compute expectations of k-fold sums over functions of the population at k different times. The result generalises [14] to the non-local setting, as introduced in [11] and [8]. As an application, we consider the case when the branching process is critical, and conditioned to survive for a large time.…”

“…Remark 7. We note that the assumptions above are inherited from [11,8], where asymptotic results concerning criticality and moment growth were considered. Whereas the Assumption 2.1 is clearly a standard criticality assumption, the remaining assumptions 2.2 -2.5 can be interpreted as follows.…”

The genealogy of a nearly critical branching processes in varying environment

“…Such a k-spine construction has also been used to extract information about the genealogy of several branching processes [18,23,10], and the current work follows a similar path. We also want to point at [13,20] for a computation of higher order moments for general spatial branching processes, and the derivation of a Yaglom law in this context.…”