  The notion of stability can be generalised to point processes by defining the
scaling operation in a randomised way: scaling a configuration by $t$
corresponds to letting such a configuration evolve according to a Markov
branching particle system for -$\log t$ time. We prove that these are the only
stochastic operations satisfying basic associativity and distributivity
properties and we thus introduce the notion of branching-stable point
processes. We characterise stable distributions with respect to local branching
as thinning-stable point processes with multiplicities given by the
quasi-stationary (or Yaglom) distribution of the branching process under
consideration. Finally we extend branching-stability to random variables with
the help of continuous branching (CB) processes, and we show that, at least in
some frameworks, $\mathcal{F}$-stable integer random variables are exactly Cox
(doubly stochastic Poisson) random variables driven by corresponding CB-stable
continuous random variables.
