Dimension drop for finitely supported random walks in PSL_2(R).

Consider two hyperbolic isometries of the hyperbolic plane with no common fixed point. It is well known that if their translation lengths are big enough they generate a Schottky group. In that case, fix a probability measure on those generators and their inverses, and consider the associated random walk starting at any point of the hyperbolic plane. Almost every trajectory converges to a point in the limit set and its distribution is known as the hitting, or stationary  measure of the random walk. We will prove that this measure has Hausdorff dimension strictly less than the boundary itself. In particular, it is singular with respect to the Hausdorff measure of the limit set. This 'property' is known as dimension drop and is conjectured to happen in more general contexts. In this talk I will  present the example mentioned above, some generalizations and some difficulties that appear in other cases.