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Brownian Motion on Compact Manifolds: Cover Time and Late Points


 
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1. Title Title of document Brownian Motion on Compact Manifolds: Cover Time and Late Points
 
2. Creator Author's name, affiliation, country Amir Dembo; Stanford University
 
2. Creator Author's name, affiliation, country Yuval Peres; University of California, Berkeley
 
2. Creator Author's name, affiliation, country Jay Rosen; College of Staten Island, CUNY
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Brownian motion, manifold, cover time, Wiener sausage.
 
3. Subject Subject classification 60J65
 
4. Description Abstract Let $M$ be a smooth, compact, connected Riemannian manifold of dimension $d>2$ 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 $\limsup_{r\to 0} T(x,r)/ (r^{2-d}|\log r|) = \alpha >0$.
 
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7. Date (YYYY-MM-DD) 2003-08-25
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/139
 
10. Identifier Digital Object Identifier 10.1214/EJP.v8-139
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 8
 
12. Language English=en
 
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