Curvature conditions on Riemannian manifolds with Brownian harmonicity properties

Author:
H. R. Hughes

Journal:
Trans. Amer. Math. Soc. **347** (1995), 339-361

MSC:
Primary 58G32; Secondary 53C21, 60J65

DOI:
https://doi.org/10.1090/S0002-9947-1995-1276934-8

MathSciNet review:
1276934

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Abstract: The time and place that Brownian motion on a Riemannian manifold first exits a normal ball of radius is considered and a general procedure is given for computing asymptotic expansions, as 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

DOI:
https://doi.org/10.1090/S0002-9947-1995-1276934-8

Keywords:
Brownian motion,
harmonic space,
curvature conditions,
exit time,
exit place

Article copyright:
© Copyright 1995
American Mathematical Society