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Branching Random Walks and Martingales

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Branching Random Walks

Part of the book series: Lecture Notes in Mathematics ((LNMECOLE,volume 2151))

Abstract

The Galton–Watson branching process counts the number of particles in each generation of a branching process. In this chapter, we produce an extension, in the spatial sense, by associating each individual of the branching process with a random variable. This results in a branching random walk. We present several martingales that are naturally related to the branching random walk, and study some elementary properties.

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Notes

  1. 1.

    If \(\mathbf{P}\{\sum _{\vert x\vert =1}\mathbf{1}_{\{V (x)>0\}} > 0\} > 0\) , then the condition that ψ(t) ≤ 0 for some t > 0 is also necessary to have \(\inf _{\vert x\vert =n}V (x) \rightarrow \infty \), P ∗ -a.s. See Biggins [52].

  2. 2.

    It is always possible, by means of a simple translation, to make a branching random walk satisfy ψ(1) = 0 as long as \(\psi (1) < \infty \). The condition ψ′(1) ≤ 0 is more technical: It is to guarantee the existence of the forthcoming function Φ; see the paragraph below.

  3. 3.

    In Proposition 3.4 and Lemma 3.5 below, we simply say that Φ is a Laplace transform satisfying (3.1).

  4. 4.

    In fact, it is also unique, up to a multiplicative constant in the argument. See Biggins and Kyprianou [56].

  5. 5.

    In [157], Helly’s selection principle is stated for functions on a compact interval. We apply it to each of the intervals [0, n], and then conclude by a diagonal argument (i.e., taking the diagonal elements in a double array).

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Authors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Paris, France

    Zhan Shi

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  1. Zhan Shi
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Shi, Z. (2015). Branching Random Walks and Martingales. In: Branching Random Walks. Lecture Notes in Mathematics(), vol 2151. Springer, Cham. https://doi.org/10.1007/978-3-319-25372-5_3

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