Equidistribution of mass for random processes on finite-volume spaces

Let $G$ be a real Lie group, $\mathrm{\Lambda }\subseteq G$ a lattice, and $X=G\mathrm{/}\mathrm{\Lambda }$. We fix a probability measure $\mu$ on $G$ and consider the left random walk induced on $X$. It is assumed that $\mu$ is aperiodic, has a finite first moment, spans a semisimple algebraic group without compact factors, and has two non mutually singular convolution powers. We show that for every starting point $x\in X$, the $n$-th step distribution ${\mu }^n\ast {\delta }_x$ of the walk weak-$\ast$ converges toward some homogeneous probability measure on $X$.