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Journal of Theoretical Probability

, Volume 32, Issue 2, pp 608–632 | Cite as

Porosities of Mandelbrot Percolation

  • Artemi BerlinkovEmail author
  • Esa Järvenpää
Article
  • 13 Downloads

Abstract

We study porosities in the Mandelbrot percolation process using a notion of porosity that is based on the construction geometry. We show that, almost surely at almost all points with respect to the natural measure, the construction-based mean porosities of the set and the natural measure exist and are equal to each other for all parameter values outside of a countable exceptional set. As a corollary, we obtain that, almost surely at almost all points, the regular lower porosities of the set and the natural measure are equal to zero, whereas the regular upper porosities reach their maximum values.

Keywords

Random sets Porosity Mean porosity 

Mathematics Subject Classification (2010)

28A80 37A50 60D05 60J80 

Notes

Acknowledgements

We thank Maarit Järvenpää for interesting discussions and many useful comments. We are also thankful to the referee for the comments and suggestions on improvement in this paper.

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Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat GanIsrael
  2. 2.University ITMOSt. PetersburgRussian Federation
  3. 3.Department of Mathematical SciencesUniversity of OuluOuluFinland

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