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Additive Lévy Processes: Capacity and Hausdorff Dimension

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Fractal Geometry and Stochastics III

Part of the book series: Progress in Probability ((PRPR,volume 57))

Abstract

This is a survey on recently-developed potential theory of additive Lévy processes and its applications to fractal geometry of Lévy processes.

Additive Lévy processes arise naturally in the studies of the Brownian sheet, intersections of Lévy processes and so on. We first summarize some recent results on the novel connections between an additive Lévy process X in Rd, and a natural class of energy forms and their corresponding capacities. We then apply these results to study the Hausdorff dimension of the range and self-intersections of an ordinary Lévy process, solving several long-standing problems in the folklore of the theory of Lévy processes. We also list several open problems in this area.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Utah, 155 South 1400 East JWB 233, Salt Lake City, UT, 84112-0090, USA

    Davar Khoshnevisan

  2. Department of Statistics and Probability, Michigan State University, A-413 Wells Hall, East Lansing, 48824, MI, USA

    Yimin Xiao

Authors
  1. Davar Khoshnevisan
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  2. Yimin Xiao
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Editors and Affiliations

  1. Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität, Greifswald, 17487, Germany

    Christoph Bandt

  2. Department of Physics, University of Rome La Sapienza, Via G. Boni 20, Roma, 00162, Italy

    Umberto Mosco

  3. Mathematisches Institut, Friedrich-Schiller-Universität, Jena, 07740, Germany

    Martina Zähle

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Khoshnevisan, D., Xiao, Y. (2004). Additive Lévy Processes: Capacity and Hausdorff Dimension. In: Bandt, C., Mosco, U., Zähle, M. (eds) Fractal Geometry and Stochastics III. Progress in Probability, vol 57. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7891-3_10

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  • DOI: https://doi.org/10.1007/978-3-0348-7891-3_10

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9612-2

  • Online ISBN: 978-3-0348-7891-3

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