In statistical physics phase space behavior of an ensemble of non interacting particles can be described by the Liouville equation. In the stationary case with inflow boundary conditions on a (finite) slab the method of characteristics provides solutions with jump type discontinuities. The goal of this work was to overcome the uniqueness issues using a vanishing viscosity method. Since existing results cannot handle problems with non symmetric, parameter dependent collision operators, these approaches are extended. In particular existence of an unique solution to the parabolic-elliptic degenerated diffusive version of the stationary Liouville equation is proven. Furthermore some basic properties such as smoothness and a bound by the posed boundary conditions were established. Hereby the intrinsic Krein space structure of this problem was pointed out.