  The Levy walk model is studied in the context of the anomalous heat
conduction of one dimensional systems. In this model the heat carriers execute
Levy-walks instead of normal diffusion as expected in systems where Fourier's
law holds. Here we calculate exactly the average heat current, the large
deviation function of its fluctuations and the temperature profile of the
Levy-walk model maintained in a steady state by contact with two heat baths
(the open geometry). We find that the current is non-locally connected to the
temperature gradient. As observed in recent simulations of mechanical models,
all the cumulants of the current fluctuations have the same system-size
dependence in the open geometry. For the ring geometry, we argue that a size
dependent cut-off time is necessary for the Levy walk model to behave as
mechanical models. This modification does not affect the results on transport
in the open geometry for large enough system sizes.
