Stochastic solutions to evolution equations of non-local branching processes

We give a probabilistic representation for the solution to a nonlinear evolution equation induced by a measure-valued branching process. We first construct the involved branching processes on the set of all finite configurations of a given set, with a killing rate induced by a continuous additive functional, and with a non-local branching procedure given by a sequence of Markovian kernels. The main application is to prove stochastic aspects for a nonlinear evolution equation related to the Neumann problem and the surface measure on the boundary, which corresponds to the reflecting Brownian motion as base movement, taking the killing rate given by the local time on the boundary. We use specific potential theoretical tools.