Branching process representation for Burgers-like nonlinear partial differential equations

Abstract

We show that a large class of first-order conservation PDEs can be probabilistically represented using multi-type branching processes. The representation holds when the initial conditions are linear combinations of negative exponentials. This class contains Burgers' equation in 1D, which may develop a gradient blow-up in finite time, as well as other equations with more general nonlinearities. We also show how the time of gradient blow up can be identified by studying criticality conditions of the corresponding branching processes.