Infinitely ramified point measures and branching Lévy processes

Abstract

We call a random point measure infinitely ramified if for every n ∈ N, it has the same distribution as the n-th generation of some branching random walk. On the other hand, branching Lévy processes model the evolution of a population in continuous time, such that individuals move in space independently, according to some Lévy process, and further beget progenies according to some Poissonian dynamics, possibly on an everywhere dense set of times. Our main result connects these two classes of processes much in the same way as in the case of infinitely divisible distributions and Lévy processes: the value at time 1 of a branching Lévy process is an infinitely ramified point measure, and conversely, any infinitely ramified point measure can be obtained as the value at time 1 of some branching Lévy process.

Biggins’ martingale convergence for branching Lévy processes

“…The purpose of this work is to present a version of Biggins' martingale convergence theorem for branching Lévy processes, a family of branching processes in continuous time that was recently introduced in [BM17]. Branching Lévy processes bear the same relation to branching random walks as Lévy processes do to random walks: a branching Lévy process (Z t ) t≥0 is a point-measure valued process such that for every r > 0, its discrete-time skeleton (Z nr ) n≥0 is a branching random walk.…”

“…We call κ the cumulant generating function of Z 1 ; to justify the terminology, recall from Theorem 1.1(ii) in [BM17] that for all t ≥ 0, we have E ( Z t , e z ) = exp (tκ(z)) .…”

“…In this section, we start by summarizing the construction of the branching Lévy process with characteristics (σ 2 , a, Λ) as a particle system, referring to Sections 4 and 5 in [BM17] for a detailed account. We shall then present a version of the spinal decomposition tailored for the purpose of this proof, and finally adapt the approach of Lyons [Lyo97] to establish Theorem 1.1 and Proposition 1.4.…”

A result on power moments of Lévy-type perpetuities and its application to the Lp -convergence of Biggins’ martingales in branching Lévy processes

“…Second, we apply this criterion and the aforementioned result from [18] to derive necessary and sufficient conditions for the a.s. and the L p -convergence for p ≥ 1 of the Biggins martingale in the branching Lévy process. Thus, we obtain final versions of Theorem 1.1 and Proposition 1.4 in [9] which was our primary motivation.…”

“…The branching property of the branching Lévy process tells us that conditionally on the positions of the particles at time t the processes initiated by these particles are i.i.d. branching Lévy processes, shifted by the position of their ancestor, see [9,Fact (B)]. The branching property in combination with (4.4) imply that the process W := (W t ) t≥0 defined by…”