Application of the von Mises-Fisher distribution to Random Walk on Spheres method for solving high-dimensional diffusion-advection-reaction equations

We suggest a new efficient and reliable random walk method, continuous both in space and time, for solving high-dimensional diffusion-advection-reaction equations. It is based on a discovered intrinsic relation between the von Mises-Fisher distribution on a sphere with this type of equations. It can be formulated as follows: the von Mises-Fisher distribution uniquely defines the solution of a diffusion-advection equation in any bounded or unbounded domain if the relevant boundary value problem for this equation satisfies regular existence and uniqueness conditions. Both two- and three-dimensional transient equations are included in our considerations. The accuracy and the cost of the suggested random walk on spheres method are estimated.