Sub- and super-diffusion on Cantor sets: Beyond the paradox

There is no way to build a nontrivial Markov process having continuous trajectories on a totally disconnected fractal embedded in the Euclidean space. Accordingly, in order to delineate the diffusion process on the totally disconnected fractal, one needs to relax the continuum requirement. Consequently, a diffusion process depends on how the continuum requirement is handled. This explains the emergence of different types of anomalous diffusion on the same totally disconnected set. In this regard, we argue that the number of effective spatial degrees of freedom of a random walker on the totally disconnected Cantor set is equal to n_sp=[D]+1, where [D] is the integer part of the Hausdorff dimension of the Cantor set. Conversely, the number of effective dynamical degrees of freedom (d_s) depends on the definition of a Markov process on the totally disconnected Cantor set embedded in the Euclidean space Eⁿ (n≥n_sp). This allows us to deduce the equation of diffusion by employing the local differential operators on the F^α-support. The exact solutions of this equation are obtained on the middle-ϵ Cantor sets for different kinds of the Markovian random processes. The relation of our findings to physical phenomena observed in complex systems is highlighted.