Codice  QDD217 
Titolo  A generating function approach to branching random walks 
Data  20151126 
Autore/i  Bertacchi, D.; Zucca, F. 
Link  Download full text 
Abstract  It is well known that the behaviour of a branching process is completely described by the generating function of the offspring law and its fixed points. Branching random walks are a natural generalization of branching processes: a branching process can be seen as a onedimensional
branching random walk. We define a multidimensional generating function associated to a given branching random walk. The present paper investigates the similarities and the differences of the generating functions, their fixed points and the implications on the underlying stochastic process,
between the onedimensional (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 affected by local modications of the process. As a corollary, we describe a general procedure by which we can modify a continuoustime branching random walk which has a weak
phase and turn it into a continuoustime branching random walk which has strong local survival for large or small values of the parameter and nonstrong local survival for intermediate values of the parameter. 
