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On the Macroscopic Fractal Geometry of Some Random Sets

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Stochastic Analysis and Related Topics

Part of the book series: Progress in Probability ((PRPR,volume 72))

Abstract

This paper is concerned mainly with the macroscopic fractal behavior of various random sets that arise in modern and classical probability theory. Among other things, it is shown here that the macroscopic behavior of Boolean coverage processes is analogous to the microscopic structure of the Mandelbrot fractal percolation. Other, more technically challenging, results of this paper include:

  1. (i)

    The computation of the macroscopic Minkowski dimension of the graph of a large family of Lévy processes; and

  2. (ii)

    The determination of the macroscopic monofractality of the extreme values of symmetric stable processes.

As a consequence of (i), it will be shown that the macroscopic fractal dimension of the graph of Brownian motion differs from its microscopic fractal dimension. Thus, there can be no scaling argument that allows one to deduce the macroscopic geometry from the microscopic. Item (ii) extends the recent work of Khoshnevisan et al. (Ann Probab, to appear) on the extreme values of Brownian motion, using a different method.

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Notes

  1. 1.

    The same argument shows that if X and Y are independent subordinators, then we have the change-of-variables formula,

    $$\displaystyle{\int _{0}^{\infty }\frac{\mathrm{U}_{_{X}}(\mathrm{d}x)} {\Phi _{_{Y }}(x)} =\int _{ 0}^{\infty }\frac{\mathrm{U}_{_{Y }}(\mathrm{d}y)} {\Phi _{_{X}}(y)}.}$$

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Acknowledgements

Research supported in part by the National Science Foundation grant DMS-1307470, DMS-1608575 and DMS-1607089.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Utah, 155 South 1400 East, JWB 102, Salt Lake City, UT, 84112-0090, USA

    Davar Khoshnevisan

  2. Department of Statistics & Probability, Michigan State University, East Lansing, MI, 48824, USA

    Yimin Xiao

Authors
  1. Davar Khoshnevisan
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  2. Yimin Xiao
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Corresponding author

Correspondence to Davar Khoshnevisan .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, Connecticut, USA

    Fabrice Baudoin

  2. Department of Mathematics, Purdue University, West Lafayette, Indiana, USA

    Jonathon Peterson

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Khoshnevisan, D., Xiao, Y. (2017). On the Macroscopic Fractal Geometry of Some Random Sets. In: Baudoin, F., Peterson, J. (eds) Stochastic Analysis and Related Topics. Progress in Probability, vol 72. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-59671-6_9

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