Solving Multidimensional Fractional Fokker--Planck Equations via Unbiased Density Formulas for Anomalous Diffusion Processes

Abstract

Fractional Fokker--Planck equations are generalizations of ordinary Fokker--Planck equations that include fractional derivative operators to encapsulate trapping effects in the anomalously diffusing population they describe. They do not admit analytical solutions in general and there are significant drawbacks to current numerical approximation methods. Using Malliavin calculus, we establish formulas in the form of a mathematical expectation for the probability density functions of anomalous diffusion processes, which are known to solve fractional Fokker--Planck equations. These density formulas lend themselves easily to computation via Monte Carlo sampling and are easy to implement. Moreover, they have the capability to deal with a general model of anomalous diffusion, including the case where the parent process of the anomalous diffusion has a Levy jump component. The numerical solutions obtained are inherently unbiased, which allows them to form the basis for reliable inferences about the population dyn...