  The aim of this paper is the study of the strong local survival property for
discrete-time and continuous-time branching random walks. We study this
property by means of an infinite dimensional generating function G and a
maximum principle which, we prove, is satisfied by every fixed point of G. We
give results about the existence of a strong local survival regime and we prove
that, unlike local and global survival, in continuous time, strong local
survival is not a monotone property in the general case (though it is monotone
if the branching random walk is quasi transitive). We provide an example of an
irreducible branching random walk where the strong local property depends on
the starting site of the process. By means of other counterexamples we show
that the existence of a pure global phase is not equivalent to nonamenability
of the process, and that even an irreducible branching random walk with the
same branching law at each site may exhibit non-strong local survival. Finally
we show that the generating function of a irreducible BRW can have more than
two fixed points; this disproves a previously known result.
