We begin with a quick review of the Feynman-Kac equations.  These allow one to represent the solution of linear elliptic and parabolic partial differential equations (PDEs) as expected values over stochastic processes.  The particular stochastic process for a given PDE is the solution to a stochastic differential equation defined via the elliptic operator in the PDE.  We then briefly discuss methods for nonlinear parabolic equations known as reaction-diffusion equations.

We then return to elliptic PDEs and discuss, in detail, several acceleration techniques that are widely applicable Monte Carlo methods.  We begin with the "walk on spheres" algorithm, followed by the the "Greens function first-passage" method, the "simulation-tabulation" method, "last passage" methods, the "walk on the boundary" method, and finally the "walk on subdomains" method.  These various Monte Carlo methods are presented within the context of various problems that arise in flow through porous media, electrostatics, and continuum biochemistry.

We also present the example of the "telegrapher's" equation, an hyperbolic equation, as solved stochastically, and some nonlinear PDE examples.