The Annals of Probability

Hitting Distributions of Small Geodesic Spheres

Ming Liao

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Abstract

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.

Article information

Source
Ann. Probab., Volume 16, Number 3 (1988), 1039-1050.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991676

Digital Object Identifier
doi:10.1214/aop/1176991676

Mathematical Reviews number (MathSciNet)
MR942754

Zentralblatt MATH identifier
0651.58037

JSTOR
links.jstor.org

Subjects
Primary: 58G32

Keywords
Riemannian manifolds Brownian motion geodesic spheres hitting distributions hitting times scalar curvature Ricci curvature

Citation

Liao, Ming. Hitting Distributions of Small Geodesic Spheres. Ann. Probab. 16 (1988), no. 3, 1039--1050. doi:10.1214/aop/1176991676. https://projecteuclid.org/euclid.aop/1176991676


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