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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Curvature conditions on Riemannian manifolds with Brownian harmonicity properties
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by H. R. Hughes PDF
Trans. Amer. Math. Soc. 347 (1995), 339-361 Request permission


The time and place that Brownian motion on a Riemannian manifold first exits a normal ball of radius $\varepsilon$ is considered and a general procedure is given for computing asymptotic expansions, as $\varepsilon$ decreases to zero, for joint moments of the first exit time and place random variables. It is proven that asymptotic versions of exit time and place distribution properties that hold on harmonic spaces are equivalent to certain curvature conditions for harmonic spaces. In particular, it is proven that an asymptotic mean value condition involving first exit place is equivalent to certain levels of curvature conditions for harmonic spaces depending on the order of the asymptotics. Also, it is proven that an asymptotic uncorrelated condition for first exit time and place is equivalent to weaker curvature conditions at corresponding orders of asymptotics.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 339-361
  • MSC: Primary 58G32; Secondary 53C21, 60J65
  • DOI:
  • MathSciNet review: 1276934