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Branching random walks. École d’Été de Probabilités de Saint-Flour XLII – 2012. (English) Zbl 1348.60004

Lecture Notes in Mathematics. 2151. École d’Été de Probabilités de Saint-Flour. Cham: Springer (ISBN 978-3-319-25371-8/pbk; 978-3-319-25372-5/ebook). x, 133 p. (2015).
The lecture notes under review provide an introduction to supercritical branching random walks (BRW). The author focuses on the discrete-time setting in one spatial dimension. The BRW is defined as follows. At time \(0\), it starts with a single ancestor particle located at the origin. At time one, this particle is replaced by a family of children, whose positions form some point process \(\Xi\) on \(\mathbb R\) with finitely many points. More generally, at any time \(n\), every particle in the system is replaced by a family of offspring, whose displacements with respect to the position of the parent form an independent realization of the point process \(\Xi\). It is assumed that the expected number of points in \(\Xi\) is \(>1\), i.e., the BRW is supercritical.
The most deep results of this book, proved in Chapter 5, deal with the asymptotic properties of the position \(M_n\) of the left-most particle in the BRW, a question on which a lot of progress has been done in recent years. Under suitable assumptions on the BRW (which, in particular, guarantee that the speed of linear growth of \(M_n\) is \(0\)), it is shown that for all \(u\in\mathbb R\), \[ \lim_{n\to\infty} \mathbb P\left[M_n - \frac 32 \log n > u \right] = \mathbb E [e^{-C e^{u}D_\infty}], \] where \(D_\infty\) is the limit of the derivative martingale associated with the BRW. This result is due to E. Aïdékon [Ann. Probab. 41, No. 3A, 1362–1426 (2013; Zbl 1285.60086)]. Various martingales associated with the BRW (including the derivative martingale) are studied in Chapter 3. The proofs exploit the spinal structure of the BRW which is introduced in Chapter 2 in the simpler setting of Galton-Watson trees and exploited in full generality in Chapter 4.
In the last Chapters 6 and 7, the author provides a short survey on some modifications of the BRW, including the BRW with absorption, the \(N\)-BRW (in which in any generation only \(N\) particles with smallest spatial positions survive), and BRW on Galton-Watson trees.
These nice lecture notes introduce the reader into deep results on branching random walks obtained in the recent few years. The book will be useful to all specialists in probability theory.

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G50 Sums of independent random variables; random walks
60G42 Martingales with discrete parameter
60G70 Extreme value theory; extremal stochastic processes

Citations:

Zbl 1285.60086
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