Anomalous diffusion on a fractal mesh

An exact analytical analysis of anomalous diffusion on a fractal mesh is presented. The fractal mesh structure is a direct product of two fractal sets which belong to a main branch of backbones and side branch of fingers. The fractal sets of both backbones and fingers are constructed on the entire (infinite) $y$ and $x$ axises. To this end we suggested a special algorithm of this special construction. The transport properties of the fractal mesh is studied, in particular, subdiffusion along the backbones is obtained with the dispersion relation $⟨x^2(t)⟩\sim t^{\beta }$, where the transport exponent $\beta <1$ is determined by the fractal dimensions of both backbone and fingers. Superdiffusion with $\beta >1$ has been observed as well when the environment is controlled by means of a memory kernel.