  A Levy walk is a non-Markovian stochastic process in which the elementary
steps of the walker consist of motion with constant speed in randomly chosen
directions and for a random period of time. The time of flight is chosen from a
long-tailed distribution with a finite mean but an infinite variance. Here we
consider an open system with boundary injection and removal of particles, at
prescribed rates, and study the steady state properties of the system. In
particular, we compute density profiles, current and current fluctuations in
this system. We also consider the case of a finite density of Levy walkers on
the ring geometry. Here we introduce a size dependent cut-off in the time of
flight distribution and consider properties of current fluctuations.
