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Probability Theory and Related Fields

, Volume 140, Issue 1–2, pp 129–167 | Cite as

Hausdorff dimension of the contours of symmetric additive Lévy processes

  • Davar Khoshnevisan
  • Narn-Rueih Shieh
  • Yimin Xiao
Article

Abstract

Let X 1, ..., X N denote N independent, symmetric Lévy processes on R d . The corresponding additive Lévy process is defined as the following N-parameter random field on R d : \({\mathfrak{X}(t) :=X_1(t_1) + \cdots + X_N(t_N)\quad(t\in {\bf R}^N_+).}\)

Khoshnevisan and Xiao (Ann Probab 30(1):62–100, 2002) have found a necessary and sufficient condition for the zero-set \({\mathfrak{X}^{-1}(\{0\})}\) of \({\mathfrak{X}}\) to be non-trivial with positive probability. They also provide bounds for the Hausdorff dimension of \({\mathfrak{X}^{-1}(\{0\})}\) which hold with positive probability in the case that \({\mathfrak{X}^{-1}(\{0\})}\) can be non-void.

Here we prove that the Hausdorff dimension of \({\mathfrak{X}^{-1}(\{0\})}\) is a constant almost surely on the event \({\{\mathfrak{X}^{-1}(\{0\})\neq\varnothing\}}\) . Moreover, we derive a formula for the said constant. This portion of our work extends the well known formulas of Horowitz (Israel J Math 6:176–182, 1968) and Hawkes (J Lond Math Soc 8:517–525, 1974) both of which hold for one-parameter Lévy processes.

More generally, we prove that for every nonrandom Borel set F in (0,∞) N , the Hausdorff dimension of \({\mathfrak{X}^{-1}(\{0\})\cap F}\) is a constant almost surely on the event \({\{\mathfrak{X}^{-1}(\{0\})\cap F\neq\varnothing\}}\) . This constant is computed explicitly in many cases.

Keywords

Additive Lévy processes Level sets Hausdorff dimension 

Mathematics Subject Classification (2000)

60G70 60F15 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Davar Khoshnevisan
    • 1
  • Narn-Rueih Shieh
    • 2
  • Yimin Xiao
    • 3
  1. 1.Department of MathematicsThe University of UtahSalt Lake CityUSA
  2. 2.Department of MathematicsNational Taiwan UniversityTaipeiTaiwan
  3. 3.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA

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