  In this article, the continuous time random walk on the circle is studied. We
derive the corresponding generalized master equation and discuss the effects of
topology, especially important when Levy flights are allowed. Then, we work out
the fluid limit equation, formulated in terms of the periodic version of the
fractional Riemann-Liouville operators, for which we provide explicit
expressions. Finally, we compute the propagator in some simple cases. The
analysis presented herein should be relevant when investigating anomalous
transport phenomena in systems with periodic dimensions.
