Rigidity and equidistribution of random walks by diffeomorphisms near the conservative regime

We consider a random walk on a closed manifold $M$ driven by a probability measure $\mu$ on the space of $C^2$ diffeomorphisms. Provided $\mu$ has compact support, satisfies certain gap and pinching conditions, and is weak-$\ast$ close to a volume-preserving measure, we prove that $M$ carries a unique atom-free stationary probability measure ${\mathrm{{\rm Y}}}_{\mu }$. This measure has full Frostman dimension and coincides with volume in the volume-preserving setting. Moreover, for every $x\in M$, the $n$-step distribution ${\mu }^{\ast n}\ast {\delta }_x$ converges to ${\mathrm{{\rm Y}}}_{\mu }$ unless $x$ is contained in a finite $\mu$-invariant set. Our result applies to a variety of situations, including bi-expanding random walks on surfaces, non-linear perturbations of Zariski-dense random walks on the torus ${\mathbb{T}}^d$, on cocompact lattice quotients of $\mathrm{S}\mathrm{O}(2,1)$ and $\mathrm{S}\mathrm{O}(3,1)$, and on the sphere ${\mathbb{S}}^d$.