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The limiting angle of certain riemannian brownian motions
Author(s) -
Hsu Pei,
March Peter
Publication year - 1985
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160380608
Subject(s) - mathematics , geodesic , combinatorics , riemannian manifold , conjecture , brownian motion , sectional curvature , invariant (physics) , orthogonal complement , infinity , curvature , mathematical analysis , geometry , scalar curvature , mathematical physics , statistics , subspace topology
Let M be a Cartan‐Hadamard manifold of dimension d ≧ 3, let p ϵ M and x = exp {r(x)θ(x)} be geodesic polar coordinates with pole p and let X be the Brownian motion on M . Let Sect M (x) denote the sectional curvature of any plane section in M x . We prove that for each c > 2, there is a 0 < β < 1 such that if ‐ L 2 r(x) 2β ≦ Sect M (x) ≦ ‐cr(x) −2 for all x in the complement of a compact set, then lim t → ∞ θ( X t ) exists a.s. and defines a nontrivial invariant random variable. The Dirichlet problem at infinity and a conjecture of Greene and Wu are also discussed.