Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus

Let $\nu$ be a probability measure on $\mathrm{SL}_d(\mathbb{Z})$ satisfying the moment condition $\mathbb{E}_\nu(\Vert g\Vert^\epsilon)<\infty$ for some $\epsilon$. We show that if the group generated by the support of $\nu$ is large enough, in particular if this group is Zariski dense in $\mathrm{SL}_d$, for any irrational $x \in \mathbb{T}^d$ the probability measures $\nu^{* n} * \delta_x$ tend to the uniform measure on $\mathbb{T}^d$. If in addition $x$ is Diophantine generic, we show this convergence is exponentially fast.