Ergodic cocycles in hyperbolic and Hadamard spaces

We consider discrete random dynamical systems induced by a non-elementary group action on a non-proper hyperbolic space. We prove that if the system is ergodic and satisfies the ``asymptotic past and future independence condition’’ as defined by Bader and Furman, the associated ergodic cocycle converges to the Gromov boundary almost surely. If the cocycle has finite first moment, we show that its drift is positive. Using hyperbolic models introduced by Petyt-Spriano-Zalloum, we prove analogous statements for groups acting on Hadamard spaces with a pair of contracting elements.