## Electronic Journal of Probability

### Brownian Motion on Compact Manifolds: Cover Time and Late Points

#### Abstract

Let $M$ be a smooth, compact, connected Riemannian manifold of dimension $d \gt 2$ and without boundary. Denote by $T(x,r)$ the hitting time of the ball of radius $r$ centered at $x$ by Brownian motion on $M$. Then, $C_r(M)=\sup_{x \in M} T(x,r)$ is the time it takes Brownian motion to come within $r$ of all points in $M$. We prove that $C_r(M)/(r^{2-d}|\log r|)$ tends to $\gamma_d V(M)$ almost surely as $r\to 0$, where $V(M)$ is the Riemannian volume of $M$. We also obtain the multi-fractal spectrum'' $f(\alpha)$ for late points'', i.e., the dimension of the set of $\alpha$-late points $x$ in $M$ for which $\limsup_{r\to 0} T(x,r)/ (r^{2-d}|\log r|) = \alpha \gt 0$.

#### Article information

Source
Electron. J. Probab., Volume 8 (2003), paper no. 15, 14 p.

Dates
First available in Project Euclid: 23 May 2016

https://projecteuclid.org/euclid.ejp/1464037588

Digital Object Identifier
doi:10.1214/EJP.v8-139

Mathematical Reviews number (MathSciNet)
MR1998762

Zentralblatt MATH identifier
1063.58021

Subjects

#### Citation

Dembo, Amir; Peres, Yuval; Rosen, Jay. Brownian Motion on Compact Manifolds: Cover Time and Late Points. Electron. J. Probab. 8 (2003), paper no. 15, 14 p. doi:10.1214/EJP.v8-139. https://projecteuclid.org/euclid.ejp/1464037588

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