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July, 1988 Hitting Distributions of Small Geodesic Spheres
Ming Liao
Ann. Probab. 16(3): 1039-1050 (July, 1988). DOI: 10.1214/aop/1176991676


Let $M$ be an $n$-dimensional Riemannian manifold, $m \in M$ and $T$ be the hitting time of an $r$-sphere around $m$ by Brownian motion $X_t$. We have, for any smooth function $g$ on the unit sphere $S$, under normal coordinates, $E^m \lbrack g(X_T/r) \rbrack = Ig + r^2I(\nu_2g) + r^3 I(\nu_3g) + O(r^4)$ and $E^m \lbrack Tg(X_T/r) \rbrack = E^m \lbrack T \rbrack E^m \lbrack g(X_T/r) \rbrack + r^5c \sum_i \partial_i sI(z_i g) + O(r^6)$, where $I$ is the uniform probability distribution on $S, \nu_2$ and $\nu_3$ are smooth functions on $S$ whose expressions involve scalar curvature, Ricci curvature and their derivatives at $m, c$ is a constant and $s$ is the scalar curvature. $\nu_2 = 0$ if and only if either $n = 2$ or $M$ is an Einstein manifold.


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Ming Liao. "Hitting Distributions of Small Geodesic Spheres." Ann. Probab. 16 (3) 1039 - 1050, July, 1988.


Published: July, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0651.58037
MathSciNet: MR942754
Digital Object Identifier: 10.1214/aop/1176991676

Primary: 58G32

Keywords: Brownian motion , geodesic spheres , hitting distributions , hitting times , Ricci curvature , Riemannian manifolds , Scalar curvature

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 3 • July, 1988
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