  The epidemic-type aftershock sequence model (ETAS) is a simple stochastic
process modeling seismicity, based on the two best-established empirical laws,
the Omori law (power law decay ~1/t^{1+\theta} of seismicity after an
earthquake) and Gutenberg-Richter law (power law distribution of earthquake
energies). In order to describe also the space distribution of seismicity, we
use in addition a power law distribution ~1/r^{1+\mu} of distances between
triggered and triggering earthquakes. We present an exact mapping between the
ETAS model and a class of CTRW (continuous time random walk) models, based on
the identification of their corresponding Master equations. This mapping allows
us to use the wealth of results previously obtained on anomalous diffusion of
CTRW. We provide a classification of the different regimes of diffusion of
seismic activity triggered by a mainshock. Specifically, we derive the relation
between the average distance between aftershocks and the mainshock as a
function of the time from the mainshock and of the joint probability
distribution of the times and locations of the aftershocks. Our predictions are
checked by careful numerical simulations. We stress the distinction between the
``bare'' Omori law describing the seismic rate activated directly by a
mainshock and the ``renormalized'' Omori law taking into account all possible
cascades from mainshocks to aftershocks of aftershock of aftershock, and so on.
In particular, we predict that seismic diffusion or sub-diffusion occurs and
should be observable only when the observed Omori exponent is less than 1,
because this signals the operation of the renormalization of the bare Omori
law, also at the origin of seismic diffusion in the ETAS model.
