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-$*$ close to a volume-preserving measure, we prove that $M$ carries a unique atom-free stationary probability measure $\Upsilon_{\mu}$. 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 $\mu^{*n} * \delta_x$ converges to $\Upsilon_{\mu}$ 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 $\mathrm{SO}(2,1)$ and $\mathrm{SO}(3,1)$, and on the sphere $\mathbb{S}^d$.