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

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.