Power Laws and Logarithmic Oscillations in Diffusion Processes on Discrete Ultrametric Spaces

It is shown that if the initial condition of the Cauchy problem for the diffusion equation on a general discrete ultrametric space is spherically symmetric with respect to some point, then this problem has an exact analytical solution. An exact solution of this problem is presented for pure ultrametric diffusion, as well as for ultrametric diffusion with a reaction sink concentrated at the center of spherical symmetry. Conditions on the ultrametric and the distribution of the number of states in ultrametric spheres are found that lead at large times to the asymptotic behavior of the solutions obtained in the form of a power law modulated by a bounded function that is log-periodic under some additional conditions.