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 Electronic Journal of Probability > Vol. 8 (2003) > Paper 15 open journal systems 


Brownian Motion on Compact Manifolds: Cover Time and Late Points

Amir Dembo, Stanford University
Yuval Peres, University of California, Berkeley
Jay Rosen, College of Staten Island, CUNY


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 $ep rar 0$, where $V(M)$ is the Riemannian volume of $M$. We also obtain the ``multi-fractal spectrum'' $f(alpha)$ for ``late points'',


Full text: PDF

Pages: 1-14

Published on: August 25, 2003
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Electronic Journal of Probability. ISSN: 1083-6489