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- Title
Hausdorff dimension of the contours of symmetric additive Lévy processes.
- Authors
Khoshnevisan, Davar; Narn-Rueih Shieh; Yimin Xiao
- 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.
- Subjects
PROBABILITY theory; HAUSDORFF measures; MEASURE theory; LEVY processes; MATHEMATICAL analysis; RANDOM walks
- Publication
Probability Theory & Related Fields, 2008, Vol 140, Issue 1/2, p129
- ISSN
0178-8051
- Publication type
Article
- DOI
10.1007/s00440-007-0060-7