Diffusions and Waves

There are two ways of reducing a scalar equation in an heterogeneous medium to a problem whose solution can be represented by means of path integrals with respect to an associated diffusion. One way is to take the Laplace transform in time and obtain an elliptic problem. The other is to apply a transmutation operator to obtain a parabolic problem. The path integral representation of the solution allows both theoretical analysis and Monte Carlo simulation for computational purposes. Here we do some of the first.