[339.1.1] Fractional Bochner-Levy-Riesz diffusion represents
an interesting mathematical generalization.
[339.1.2] It remains to be seen, however, whether and in which approximation
fractional spatial derivatives can arise phenomenologically
in mathematical models of an underlying physical reality that
obeys spatial locality.
[339.1.3] If such an approximation should exist for physical particle
transport processes, such that physical particle trajectories
can be identified with the mathematical paths of a Levy process,
then it seems to be an open problem how to understand the nonlocal
dependence of stationary states in these systems on the location of,
and particle concentration in distant and remote containers.
[339.1.4] It will also be interesting to understand whether and in which
sense the experiments are reproducible and how the apparatus
can be shielded or isolated from the environment.
[339.1.5] Without experimental evidence for the mathematical predictions
of fractional potential theory it seems difficult
to reconcile nonlocality in space (i.e. the case 0<α<2)
with theory and experiment.
