  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 from almost every
starting point in an affine invariant submanifold towards the associated affine
invariant measure, 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.
