Coincidence, equivalence and singularity of harmonic measures

In the absence of measures fully invariant with respect to a group action, this role can be to a certain extent played by the measures "invariant on average", with respect to a certain fixed distribution on the group. These measures are called stationary, and they naturally arise as harmonic measures of random walks. I will provide several partial answers to the general question about the dependence of stationary measures on the underlying step distributions on the group. The talk is based on joint work with Behrang Forghani.