The talk is based on joint work with Wolfgang Woess. There is a large body
of literature devoted to the quantitative aspects of branching random
walks on the additive group of real numbers and to the behaviour of the
associated martingales. In what concerns more general state spaces rich
enough to have a non-trivial topological boundary at infinity (like, for
instance, infinite trees), it is natural to ask about the limit behaviour
of the branching populations in geometric terms. Non-trivial limit sets of
random population sequences were first exhibited by Liggett (1996) and
later studied in similar situations by Hueter – Lalley (2000), Benjamini – Muller (2012),
Candellero – Roberts (2015) and Hutchcroft (2020).

We are looking at branching random walks from a different and apparently
novel angle. We are interested in the random limit boundary measures
arising from the uniform distributions on sample populations. Unlike with
the limit sets, the very existence of the limit measures is already a
non-trivial problem. We consider and solve this problem in two different
setups: in the topological one (when the boundary of the state space is
provided by a certain compactification) and in the measure-theoretical one
(when we are dealing with the Poisson or exit boundary of the underlying
Markov chain on the state space).