Anomalous recurrence of Markov chains on negatively curved manifolds

We present a recurrence-transience classification for discrete-time Markov chains on manifolds with negative curvature. Our classification depends only on geometric quantities associated to the increments of the chain, defined via the Riemannian exponential map. We deduce that a recurrent chain that has zero average drift at every point cannot be uniformly elliptic, unlike in the Euclidean case. We also give natural examples of zero-drift recurrent chains on negatively curved manifolds, including on a stochastically incomplete manifold.