On Convergence of Random Walks on Moduli Space

Roland Prohaska

Published 2021-02-08Version 1

The purpose of this note is to establish convergence of random walks on the moduli space of Abelian differentials on compact Riemann surfaces in two different modes: convergence of the $n$-step distributions towards the normalized Masur-Veech measure from almost every starting point, and almost sure pathwise equidistribution towards the affine invariant measure on the $SL_2(\mathbb{R})$-orbit closure of an arbitrary starting point. These are analogues to previous results for random walks on homogeneous spaces.