  Since its formulation in the late 1940s, the Feynman-Kac formula has proven
to be an effective tool for both theoretical reformulations and practical
simulations of differential equations. The link it establishes between such
equations and stochastic processes can be exploited to develop Monte Carlo
sampling methods that are effective, especially in high dimensions. There exist
many techniques of improving standard Monte Carlo sampling methods, a
relatively new development being the so-called Multilevel Monte Carlo method.
This paper investigates the applicability of multilevel ideas to the stochastic
representation of partial differential equations by the Feynman- Kac formula,
using the Walk on Sphere algorithm to generate the required random paths. We
focus on the Laplace equation, the simplest elliptic PDE, while mentioning some
extension possibilities.
