Abstract
Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m≥ 3 and that, outside a fixed compact set, the sectional curvatures are bounded above by −c 1/{r 2 ln r} and below by −c 2 r 2, where c 1 and c 2 are two positive constants and r is the geodesic distance from a fixed point. We show that, when κ≥ 1 satisfies certain conditions, the angular part of a κ-quasi-conformal Γ-martingale on M tends to a limit as time tends to infinity and the closure of the support of the distribution of this limit is the entire sphere at infinity. This improves both a result of Le for Brownian motion and also results concerning the non-existence of κ-quasi-conformal harmonic maps from certain types of Riemannian manifolds into M.
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Received: 19 September 1997
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Le, H. Limiting angles of Γ-martingales. Probab Theory Relat Fields 114, 85–96 (1999). https://doi.org/10.1007/s004400050222
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DOI: https://doi.org/10.1007/s004400050222