Recently I read about a Monte Carlo method useful to solve the Dirichlet problem $$(1)\left\{\begin{matrix} \Delta u(x) = 0,\quad x\in D\\ u(x) = f(x),\quad x\in\partial D \end{matrix}\right.$$ All of the method is based in the fact $$u(x) = \mathbb{E}(B_\tau)\quad (x\in D)$$ where $B_t$ is a Brownian motion starting in $x$ and $\tau=\inf\{t\geq0:B(t)\not\in D\}$.
My question is, do exist more general methods of this style for solving PDE? In particular, I am interested in methods for solving elliptic PDE based in a particular stochastic process, but similar things are also of my interest.
Thanks!
Yes!
There is the Feynman-Kac formula that represents the solution of a rather broad class of PDEs (heat equation, advection-diffusion equation, Black-Scholes PDE in log form are included) as the expectation over a random walk.
In particular, for elliptic PDEs, look here.