Integral Transforms, Anomalous Diffusion, and the Central Limit Theorem

We present new connections among anomalous diffusion (AD), normal diffusion (ND) and the Central Limit Theorem (CLT). This is done by defining new canonical Cartesian-like position and Cartesian-like momentum variables and canonically quantizing these according to Dirac to define generalized negative semi-definite and selfadjoint Laplacian operators. These lead to new generalized Fourier transformations (GFT) and associated generalized probability distributions, which are form invariant under the corresponding transform. The new Laplacians also lead us to postulate generalized diffusion equations (GDE), which imply a connection to the CLT. We show that the derived diffusion equations have the O’Shaughnessy-Procaccia equations (OPE) as a special case. 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. These tools allow us to prove the CLT for this class of diffusion equations.