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.
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The research of D. Kh. and Y. X. was supported by the United States NSF grant DMS-0404729.
The research of N.-R. S. was supported by a grant from the Taiwan NSC.
An erratum to this article is available at http://dx.doi.org/10.1007/s00440-008-0184-4.
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Khoshnevisan, D., Shieh, NR. & Xiao, Y. Hausdorff dimension of the contours of symmetric additive Lévy processes. Probab. Theory Relat. Fields 140, 129–167 (2008). https://doi.org/10.1007/s00440-007-0060-7
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DOI: https://doi.org/10.1007/s00440-007-0060-7
Keywords
- Additive Lévy processes
- Level sets
- Hausdorff dimension
Mathematics Subject Classification (2000)
- 60G70
- 60F15