Degenerate random perturbations of Anosov diffeomorphisms

This paper deals with random perturbations of diffeomorphisms on n-dimensional Riemannian manifolds with distributions supported on k-dimensional disks, where k<n. First we demonstrate general but not very intuitive conditions which guarantee that all invariant measures for rank k random perturbations of $C^2$ diffeomorphisms are absolutely continuous with respect to the Riemannian measure on M. For two subclasses of Anosov diffeomorphisms: hyperbolic toral automorphisms and Anosov diffeomorphisms with codimension 1 stable manifolds, the above conditions are modified in order to relate k-dimensional disks that support the distributions to certain foliations that arise from Anosov diffeomorphisms. We conclude that generic rank k random perturbations have absolutely continuous invariant measures.