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The aim of this paper is the study of the strong local survival property fordiscrete-time and continuous-time branching random walks. We study thisproperty by means of an infinite dimensional generating function G and amaximum principle which, we prove, is satisfied by every fixed point of G. Wegive results about the existence of a strong local survival regime and we provethat, unlike local and global survival, in continuous time, strong localsurvival is not a monotone property in the general case (though it is monotoneif the branching random walk is quasi transitive). We provide an example of anirreducible branching random walk where the strong local property depends onthe starting site of the process. By means of other counterexamples we showthat the existence of a pure global phase is not equivalent to nonamenabilityof the process, and that even an irreducible branching random walk with thesame branching law at each site may exhibit non-strong local survival. Finallywe show that the generating function of a irreducible BRW can have more thantwo fixed points; this disproves a previously known result.

Strong local survival of branching random walks is not monotone