Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method

Abstract

We study semi-linear elliptic PDEs with polynomial non-linearity in bounded domains and provide a probabilistic representation of their solution using branching diffusion processes. When the non-linearity involves the unknown function but not its derivatives, we extend previous results in the literature by showing that our probabilistic representation provides a solution to the PDE without assuming its existence. In the general case, we derive a new representation of the solution by using marked branching diffusion processes and automatic differentiation formulas to account for the non-linear gradient term. We consider several examples and estimate their solution by using the Monte Carlo method.