Kaimanovich and Masur showed that a random walk on the mapping class group for an initial distribution whose support generates a nonelementary subgroup when projected into Teichmüller space converges almost surely to a point in the space of projective measured foliations on the surface. This defines a harmonic measure on . Here, we show that when the initial distribution has finite support, the corresponding harmonic measure is singular with respect to the natural Lebesgue measure class on .
"Harmonic measures for distributions with finite support on the mapping class group are singular." Duke Math. J. 163 (2) 309 - 368, 1 February 2014. https://doi.org/10.1215/00127094-2430368