Brownian motion approach for spatial PDEs with stochastic data

We deal with PDE problems of elliptic type that are classically solved by the finite element method in the case of deterministic data.
Via the Feyman-Kac formula we give a stochastic representation providing us with
point-wise solutions. We show how to use the stochastic representation to generate a spatial approximation of the solution. Later we
extend this to the case of stochastic data and discuss the advantages
compared to FEM for such situations. Our main application will be Darcy's law.