Convergence of martingales on manifolds of negative curvature
Let \(X_ t\) be a martingale on a Riemannian manifold M with sectional curvatures bounded from above by \(-k^ 2<0.\) If \(X_ t\) moves in at least two 'random directions' with not too different never vanishing intensities and \(X_ t\) does not escape from unit balls too fast, then \(X_{\infty}=\lim_{t\to \infty}X_ t\) exists a.s. in the sphere at infinity (the set of directions of geodesic rays) (Theorem A). When \(X_ t\) is Brownian motion Theorem A gives a.s. convergence to a limit in the sphere at infinity if the sectional curvatures of M do not decay to minus infinity too fast with radial distance (Theorem B). This implies the existence of non-constant bounded harmonic functions on M, namely \(f(x)=E_ x(h(X_{\infty}))\) with \(h\in C(S^{d-1})\). For the proof of Theorem B a lower bound for the probability that Brownian motion remains in a ball of radius a up to time t is established for the sectional curvatures of M being pinched between two negative constants.
existence of non-constant bounded harmonic functions
scientific article; zbMATH DE number 3901750