Random Walk, Laplace Equation, Poisson Equation, Brownian Motion, Maxwell Boltzmann.
a PDE was already known to be applicable based on the Feynman-Kac theorem. According to the theorem, if the point where the path of the Brownian motion exits a boundary D after starting at x is defined as Exit(x,D), the solution of the PDE at x can be obtained by an expectation value of