I'm a mathematics student with zero physical intuition. In my course, we learned that Brownian motion can be used to construct the solutions to certain PDEs, including Laplace's equation and the heat equation.
For Laplace's equation, to find the solution $f(x)$ in a bounded domain $U$ with given boundary conditions on $\partial U$, you can start a Brownian motion at $x$ and take the expectation over the exit distribution (that is, the value of the boundary data at the point the BM first hits $\partial U$).
For the heat equation and similar PDE, you can use the Feynman-Kac formula.
Both these equations have physical interpretations, as the equation for a steady-state and the equation for heat. Further, you can think of Brownian motion as diffusion. In light of this (and the fact that Feynman was a physicist...) it seems there should be some physical, heuristic reasons why these solution methods work. The proofs I learned are not particularly enlightening.
What is this physical intuition for these solution formulas?