Random walks with bounded first moment on finite-volume spaces

Let $G$ be a real Lie group, $\mathrm{\Lambda }\le G$ a lattice, and $\mathrm{\Omega }=G\mathrm{/}\mathrm{\Lambda }$. We study the equidistribution properties of the left random walk on $\mathrm{\Omega }$ induced by a probability measure $\mu$ on $G$. It is assumed that $\mu$ has a finite first moment, and that the Zariski closure of the group generated by the support of $\mu$ in the adjoint representation is semisimple without compact factors. We show that for every starting point $x\in \mathrm{\Omega }$, the $\mu$-walk with origin $x$ has no escape of mass, and equidistributes in Ces`aro averages toward some homogeneous measure. This extends several fundamental results due to Benoist-Quint and Eskin-Margulis for walks with finite exponential moment.