A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees

Abstract

The solutions to a large class of non-linear parabolic PDEs are given in terms of expectations of suitable functionals of a tree of branching particles. A sufficient, and in some cases necessary, condition is given for the integrability of the stochastic representation, using a comparison scalar PDE. In cases where the representation fails to be integrable, a sequence of pruned trees is constructed, producing approximate stochastic representations that in some cases converge, globally in time, to the solution of the original PDE.