  We present new connections among anomalous diffusion (AD), normal diffusion
(ND) and the Central Limit Theorem. This is done by defining a point
transformation to a new position variable, which we postulate to be Cartesian,
motivated by considerations from super-symmetric quantum mechanics. Canonically
quantizing in the new position and momentum variables according to Dirac gives
rise to generalized negative semi-definite and self-adjoint Laplacian
operators. These lead to new generalized Fourier transformations and associated
probability distributions, which are form invariant under the corresponding
transform. The new Laplacians also lead us to generalized diffusion equations,
which imply a connection to the CLT. We show that the derived diffusion
equations capture all of the Fractal and Non-Fractal Diffusion equations of
O'Shaughnessy and Procaccia. However, we also obtain new equations that cannot
(so far as we are able to tell) be expressed as examples of the O'Shaughnessy
and Procaccia equations. These equations also possess asymptotics that are
related to a CLT but with bi-modal distributions as limits. The results show,
in part, that experimentally measuring the diffusion scaling law can determine
the point transformation (for monomial point transformations). We also show
that AD in the original, physical position is actually ND when viewed in terms
of displacements in an appropriately transformed position variable. Finally, we
show that there is a new, anomalous diffusion possible for bi-modal probability
distributions that also display attractor behavior which is the consequence of
an underlying CLT.
