scattering cross sections currently measured for example at the LHC. These quantum fluctuations can be thought of as being built up, as the hadrons are accelerated, by the successive branchings first of their constituant quarks into quark-gluon pairs, and then of the gluons into pairs of gluons, with some diffusion in their momenta. The dynamics of these gluons is actually exactly the kind of branching diffusion process that we are going to address in this work. Therefore, results that do not depend on the detailed properties of the particular branching random walk considered may be transposed to particle physics, and give quantitative insight into hadronic scattering cross sections.

Of course, the applications of branching random walks are much wider than particle physics. Branching random walks may for example generate Cayley trees which would represent the configuration space of directed polymers in random media [@DS:1988].\
Although our discussion will be very general, for definiteness, we shall consider a simple model for a branching random walk (BRW) in continuous time $t$ and one-dimensional space $x$, defined by two elementary processes: Each particle diffuses independently of the others with diffusion constant 1, and may split into two particles at rate 1, in such 