## The Annals of Probability

- Ann. Probab.
- Volume 23, Number 1 (1995), 173-177.

### Radial Part of Brownian Motion on a Riemannian Manifold

M. Liao and W. A. Zheng

#### Abstract

Let $\rho_t$ be the radial part of a Brownian motion in an $n$-dimensional Riemannian manifold $M$ starting at $x$ and let $T = T_\varepsilon$ be the first time $t$ when $\rho_t = \varepsilon$. We show that $E\lbrack \rho^2_{t\wedge T} \rbrack = nt - (1/6)S(x)t^2 + \sigma(t^2)$, as $t \downarrow 0$, where $S(x)$ is the scalar curvature. The same formula holds for $E\lbrack\rho^2_t\rbrack$ under some boundedness condition on $M$.

#### Article information

**Source**

Ann. Probab., Volume 23, Number 1 (1995), 173-177.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988382

**Digital Object Identifier**

doi:10.1214/aop/1176988382

**Mathematical Reviews number (MathSciNet)**

MR1330766

**Zentralblatt MATH identifier**

0834.58038

**JSTOR**

links.jstor.org

**Subjects**

Primary: 58G32

Secondary: 60J65: Brownian motion [See also 58J65]

**Keywords**

Brownian motion Riemannian manifolds

#### Citation

Liao, M.; Zheng, W. A. Radial Part of Brownian Motion on a Riemannian Manifold. Ann. Probab. 23 (1995), no. 1, 173--177. doi:10.1214/aop/1176988382. https://projecteuclid.org/euclid.aop/1176988382