Electronic Journal of Probability

Brownian Motion on Compact Manifolds: Cover Time and Late Points

Amir Dembo, Yuval Peres, and Jay Rosen

Full-text: Open access

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

Permanent link to this document
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
Primary: 60J65: Brownian motion [See also 58J65]

Keywords
Brownian motion manifold cover time Wiener sausage

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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References

  • Robert B. Ash. Real analysis and probability. Academic Press, New York, 1972.
  • Thierry Aubin. Nonlinear analysis on manifolds. Monge-Ampère equations, volume 252 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1982.
  • Gilles Courtois. Spectrum of manifolds with holes. J. Funct. Anal., 134(1):194-221, 1995.
  • Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni. Thick points for spatial Brownian motion: multifractal analysis of occupation measure. Ann. Probab., 28(1):1-35, 2000.
  • Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni. Thin points for Brownian motion. Ann. Inst. H. Poincaré Probab. Statist., 36(6):749-774, 2000.
  • Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni. Thick points for planar Brownian motion and the Erdos-Taylor conjecture on random walk. Acta Math., 186(2):239-270, 2001.
  • A. Dembo, Y. Peres, J. Rosen and O. Zeitouni, Cover times for Brownian motion and random walks in two dimensions, Ann. Math., to appear.
  • James Eells, Jr. and J. H. Sampson. Harmonic mappings of Riemannian manifolds. Amer. J. Math., 86:109-160, 1964.
  • P. J. Fitzsimmons and Jim Pitman. Kac's moment formula and the Feynman-Kac formula for additive functionals of a Markov process. Stochastic Process. Appl., 79(1):117-134, 1999.
  • H. Joyce and D. Preiss. On the existence of subsets of finite positive packing measure. Mathematika, 42(1):15-24, 1995.
  • Davar Khoshnevisan, Yuval Peres, and Yimin Xiao. Limsup random fractals. Electron. J. Probab., 5, paper no. 5, 24 pp., 2000.
  • Peter Matthews. Covering problems for Brownian motion on spheres. Ann. Probab., 16(1):189-199, 1988.
  • Peter Matthews. Covering problems for Markov chains. Ann. Probab., 16(3):1215-1228, 1988.
  • Steven Orey and S. James Taylor. How often on a Brownian path does the law of iterated logarithm fail? Proc. London Math. Soc. (3), 28:174-192, 1974.