In this talk I will outline links between branching process theory and the theory of non-linear partial differential equations, concentrating on the semilinear hyperbolic system I am currently studying.
The gist of the idea is simple: set up a branching process that corresponds to the PDE. Then, study of the branching process should yield insight (and indeed concrete information) into the PDE, and vice versa. The simplest link between branching processes and differential equations is well-known - conditioning on the first jump of the process and some simple manipulation yields a differential equation. Here we have a more complicated system, but the same idea holds.
Having the correspondence established we can carry out two parallel investigations of the system - one by standard analytic and numerical techniques, one by probability and simulation of the branching process. I will show how each theory helps advance the other, so the talk will touch upon a wide variety of techniques, but I will concentrate on the stochastic side. This utilises familiar conditioning and branching process arguments, large-deviation calculations, as well as more sophisticated martingale and change of measure techniques.