Fokker--Planck and Kolmogorov Backward Equations for Continuous Time Random Walk scaling limits

TL;DR

This paper proves that the scaling limits of Continuous Time Random Walks satisfy integro-differential equations similar to Fokker-Planck equations, even without assuming absolutely continuous laws, and derives backward equations with applications to anomalous diffusion.

Contribution

It extends the theory of CTRW scaling limits by deriving forward and backward equations without requiring the process to have absolutely continuous laws.

Findings

Scaling limits solve integro-differential equations similar to Fokker-Planck equations.

Backward governing equations are derived for the processes.

Examples illustrate the applicability to anomalous diffusion.

Abstract

It is proved that the distributions of scaling limits of Continuous Time Random Walks (CTRWs) solve integro-differential equations akin to Fokker-Planck Equations for diffusion processes. In contrast to previous such results, it is not assumed that the underlying process has absolutely continuous laws. Moreover, governing equations in the backward variables are derived. Three examples of anomalous diffusion processes illustrate the theory.