The Annals of Probability

Dynamical models for circle covering: Brownian motion and Poisson updating

Johan Jonasson and Jeffrey E. Steif

Full-text: Open access

Abstract

We consider two dynamical variants of Dvoretzky’s classical problem of random interval coverings of the unit circle, the latter having been completely solved by L. Shepp. In the first model, the centers of the intervals perform independent Brownian motions and in the second model, the positions of the intervals are updated according to independent Poisson processes where an interval of length is updated at rate α where α≥0 is a parameter. For the model with Brownian motions, a special case of our results is that if the length of the nth interval is c/n, then there are times at which a fixed point is not covered if and only if c<2 and there are times at which the circle is not fully covered if and only if c<3. For the Poisson updating model, we obtain analogous results with c<α and c<α+1 instead. We also compute the Hausdorff dimension of the set of exceptional times for some of these questions.

Article information

Source
Ann. Probab., Volume 36, Number 2 (2008), 739-764.

Dates
First available in Project Euclid: 29 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1204306966

Digital Object Identifier
doi:10.1214/07-AOP340

Mathematical Reviews number (MathSciNet)
MR2393996

Zentralblatt MATH identifier
1147.60063

Subjects
Primary: 60K99: None of the above, but in this section

Keywords
Circle coverings Brownian motion exceptional times Hausdorff dimension

Citation

Jonasson, Johan; Steif, Jeffrey E. Dynamical models for circle covering: Brownian motion and Poisson updating. Ann. Probab. 36 (2008), no. 2, 739--764. doi:10.1214/07-AOP340. https://projecteuclid.org/euclid.aop/1204306966


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References

  • Barral, J. and Fan, A.-H. (2005). Covering numbers of different points in Dvoretzky covering. Bull. Sci. Math. 129 275–317.
  • van den Berg, J., Meester, R. and White, D. G. (1997). Dynamic Boolean models. Stochastic Process. Appl. 69 247–257.
  • Billard, P. (1965). Séries de Fourier aléatoirement bornées, continues, uniformément convergentes. Ann. Sci. École Norm. Sup. (3) 82 131–179.
  • Dvoretzky, A. (1956). On covering a circle by randomly placed arcs. Proc. Natl. Acad. Sci. U.S.A. 42 199–203.
  • Fan, A.-H. (2002). How many intervals cover a point in Dvoretzky covering? Israel J. Math. 131 157–184.
  • Fan, A.-H. and Wu, J. (2004). On the covering by small random intervals. Ann. Inst. H. Poincaré Probab. Statist. 40 125–131.
  • Fitzsimmons, P. J. and Getoor, R. K. (1988). On the potential theory of symmetric Markov processes. Math. Ann. 281 495–512.
  • Getoor, R. K. and Sharp, M. J. (1984). Naturality, standardness, and weak duality for Markov processes. Z. Wahrsch. Verw. Gebiete 67 1–62.
  • Häggström, O., Peres, Y. and Steif, J. (1997). Dynamical percolation. Ann. Inst. H. Poincaré Probab. Statist. 33 497–528.
  • Kahane, J. P. (1959). Sur le recouvrement d’un cercle par des arcs disposés au hasard. C. R. Acad. Sci. Paris 248 184–186.
  • Kahane, J. P. (1985). Some Random Series of Functions, 2nd ed. Cambridge Univ. Press.
  • Lawler, G. F. (2005). Conformally Invariant Processes in the Plane. Amer. Math. Soc., Providence, RI.
  • Mandelbrot, B. (1972). On Dvoretzky coverings for the circle. Z. Wahrsch. Verw. Gebiete 22 158–160.
  • Mattila, P. (1995). Geometry of Sets and Measures in Euclidean Spaces. Cambridge Univ. Press.
  • Schramm, O. and Steif, J. E. (2007). Quantitative noise sensitivity and exceptional times for percolation. Ann. Math. To appear.
  • Shepp, L. A. (1972). Covering the circle with random arcs. Israel J. Math. 11 328–345.