  Let $\Gamma$ be a (convex-)cocompact group of isometries of the hyperbolic
space $\mathbb{H}^d$, let $M := \mathbb{H}^d/\Gamma$ be the associated
hyperbolic manifold, and consider a real valued potential $F$ on its unit
tangent bundle $T^1 M$. Under a natural regularity condition on $F$, we prove
that the associated $(\Gamma,F)$-Patterson-Sullivan densities are stationary
measures with exponential moment for some random walk on $\Gamma$. As a
consequence, when $M$ is a surface, the associated equilibrium state for the
geodesic flow on $T^1 M$ exhibit "Fourier decay", in the sense that a large
class of oscillatory integrals involving it satisfies power decay. It follows
that the non-wandering set of the geodesic flow on convex-cocompact hyperbolic
surfaces has positive Fourier dimension, in a sense made precise in the
appendix.
