Briefly speaking, the Feynman-Kac formula gives a way for constructing $C^2$ functions satisfying certain PDEs in the classical sense (at least, it's how it is explained in Oksendal's book that I am currently reading).
This method relies on Ito-diffusions and can be easily implemented from a numerical viewpoint, offering a PDE Monte-Carlo solver.
Again, let me point out how the theory relies on a classical-derivative viewpoint.
My question is: can this technique be extended to more general Sobolev spaces, in order to construct Monte Carlo methods for certain PDEs in a weak sense?
A good reference would suffice.
Thanks in advance for the help.