TitoloA generating function approach to branching random walks
Autore/iBertacchi, D.; Zucca, F.
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AbstractIt is well known that the behaviour of a branching process is completely described by the generating function of the o ffspring law and its fi xed points. Branching random walks are a natural generalization of branching processes: a branching process can be seen as a one-dimensional branching random walk. We defi ne a multidimensional generating function associated to a given branching random walk. The present paper investigates the similarities and the di fferences of the generating functions, their fi xed points and the implications on the underlying stochastic process, between the one-dimensional (branching process) and the multidimensional case (branching random walk). In particular, we show that the generating function of a branching random walk can have uncountably many fixed points and a fixed point may not be an extinction probability, even in the irreducible case (extinction probabilities are always fixed points). Moreover, the generating function might not be a convex function. We also study how the behaviour of a branching random walk is aff ected by local modi cations of the process. As a corollary, we describe a general procedure by which we can modify a continuous-time branching random walk which has a weak phase and turn it into a continuous-time branching random walk which has strong local survival for large or small values of the parameter and non-strong local survival for intermediate values of the parameter.