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Limiting angles of Γ-martingales
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  • Published: April 1999

Limiting angles of Γ-martingales

  • Huiling Le1 

Probability Theory and Related Fields volume 114, pages 85–96 (1999)Cite this article

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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Authors and Affiliations

  1. Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK. e-mail: lhl@maths.nott.ac.uk, , , , , , GB

    Huiling Le

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  1. Huiling Le
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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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  • Issue Date: April 1999

  • DOI: https://doi.org/10.1007/s004400050222

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  • Mathematics Subject Classification (1991): 58G32, 58E20
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