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Introduction

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Branching Random Walks

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

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Abstract

We introduce branching Brownian motion as well as the branching random walk, and present the elementary but very useful tool of the many-to-one formula. As a first application of the many-to-one formula, we deduce the asymptotic velocity of the leftmost position in the branching random walk. The chapter ends with some examples of branching random walks and more general hierarchical fields.

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Notes

  1. 1.

    Another historical reference is a series of papers by Ikeda et al. [141–143], who are interested in a general theory connecting probability with differential equations.

  2. 2.

    For notational simplification, we often write from now on \(\inf _{\vert x\vert =n}(\cdots \,)\) or \(\sum _{\vert x\vert =1}(\cdots \,)\), instead of \(\inf _{x:\,\vert x\vert =n}(\cdots \,)\) or \(\sum _{x:\,\vert x\vert =1}(\cdots \,)\), with \(\inf _{\varnothing }(\cdots \,):= \infty \) and \(\sum _{\varnothing }(\cdots \,):= 0\).

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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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Shi, Z. (2015). Introduction. In: Branching Random Walks. Lecture Notes in Mathematics(), vol 2151. Springer, Cham. https://doi.org/10.1007/978-3-319-25372-5_1

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