59.2 The Dirichlet Problem
A rigorous path integral: start a Brownian motion at a point x in an open region D, and let it run until it exits D (on the boundary) at the random time T_D. For any function f on the boundary, the expected value of f(B_{T_D}) gives a function of x which is harmonic in D, and equal to f on the boundary.
We use this to show that in dimensions 2 and higher Brownian motion is transient. It is neighborhood recurrent in dimension 2, but not in dimension 3 or higher.