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Brownian Motion on Compact Manifolds: Cover Time and Late Points

   
@article{EJP139,
	author = {Amir Dembo and Yuval Peres and Jay Rosen},
	title = {Brownian Motion on Compact Manifolds: Cover Time and Late Points},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {8},
	year = {2003},
	keywords = {Brownian motion, manifold, cover time, Wiener sausage.},
	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 $\limsup_{r\to 0} T(x,r)/ (r^{2-d}|\log r|) = \alpha >0$.},
	pages = {no. 15, 1-14},
	issn = {1083-6489},
	doi = {10.1214/EJP.v8-139},    
        url = {http://ejp.ejpecp.org/article/view/139}}