Abstract
What is in this chapter? Branching random walks are among the simplest continuous time spatial processes. Consider a system of particles that undergo branching and random motion on a countable graph (such as Zd or a homogeneous tree) according to the two following rules. A particle at x waits an exponential random time with rate λp(x, y) > 0 and then gives birth to a particle at y. p(x, y) are the transition probabilities of a Markov chain and λ > 0 is a parameter. A particle waits an exponential time with rate 1 and then dies.
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Notes and references
Branching random walks is one of the simplest spatial stochastic systems. It is possible to do explicit computations of critical values. As the next chapter illustrates this is rarely possible for other spatial stochastic processes. There are other ways to define branching random chains and there are many important questions on the subject that we did not treat here. See for instance Cox (1994) and the references there.
Theorem VI.2.1 was first proved in the particular case of trees by Madras and Schinazi (1992). The general case was proved by Schinazi (1993). Theorem VI.4.1 is due to Madras and Schinazi (1992).
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© 1999 Springer Science+Business Media New York
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Schinazi, R.B. (1999). Continuous Time Branching Random Walk. In: Classical and Spatial Stochastic Processes. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1582-0_6
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DOI: https://doi.org/10.1007/978-1-4612-1582-0_6
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7203-8
Online ISBN: 978-1-4612-1582-0
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