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


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.


Additive Lévy processes Level sets Hausdorff dimension 

Mathematics Subject Classification (2000)

60G70 60F15 


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  1. 1.
    Ayache A. and Xiao Y. (2005). Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets. J. Fourier Anal. Appl. 11: 407–439 zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Barlow, M.T., Perkins, E.: Levels at which every Brownian excursion is exceptional, Seminar on probability, XVIII, pp. 1–28 (1984)Google Scholar
  3. 3.
    Benjamini I., Häggström O., Peres Y. and Steif J.E. (2003). Which properties of a random sequence are dynamically sensitive?. Ann. Probab. 31(1): 1–34 zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bertoin J. (1996). Lévy Processes. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  5. 5.
    Blath J. and Mörters P. (2005). Thick points of super-Brownian motion. Probab. Theory Related Fields 131(4): 604–630 zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Blumenthal R.M. and Getoor R.K. (1968). Markov Processes and Potential Theory. Academic Press, New York zbMATHGoogle Scholar
  7. 7.
    Bochner S. (1955). Harmonic Analysis and the Theory of Probability. University of California Press, Berkeley and Los Angeles zbMATHGoogle Scholar
  8. 8.
    Dalang R.C. and Nualart E. (2004). Potential theory for hyperbolic SPDEs. Ann. Probab. 32(3): 2099–2148 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dembo, A., Peres, Y., Rosen, J., Zeitouni, O.: Thick points for intersections of planar sample paths. Trans. Am. Math. Soc. 354(12), 4969–5003 (electronic) (2002)Google Scholar
  10. 10.
    Dembo, A., Peres, Y., Rosen, J., Zeitouni, O.: Thick points for transient symmetric stable processes. Electron. J. Probab. 4(10) , 13 pp (electronic) (1999)Google Scholar
  11. 11.
    Hawkes J. (1974). Local times and zero sets for processes with infinitely divisible distributions. J.~Lond. Math. Soc. 8: 517–525 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Horowitz J. (1968). The Hausdorff dimension of the sample path of a subordinator. Israel J.~Math. 6: 176–182 zbMATHMathSciNetGoogle Scholar
  13. 13.
    Kahane J.-P. (1985). Some Random Series of Functions. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  14. 14.
    Khoshnevisan, D.: The codimension of the zeros of a stable process in random scenery, Séminaire de Probabilités XXXVII, pp. 236–245 (2003)Google Scholar
  15. 15.
    Khoshnevisan D. (2002). Multiparameter Processes. Springer, New York zbMATHGoogle Scholar
  16. 16.
    Khoshnevisan, D., Levin, D.A., Méndez-Hernández, P.J.: Exceptional times and invariance for dynamical random walks, Probab. Theory Rel. Fields, to appear (2005a)Google Scholar
  17. 17.
    Khoshnevisan D., Levin D.A. and Méndez-Hernández P.J. (2005). On dynamical Gaussian random walks. Ann. Probab. 33(4): 1452–1478 zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Khoshnevisan, D., Peres, Y., Xiao, Y.: Limsup random fractals. Electron. J. Probab. 5(5), 24 pp. (electronic) (2000)Google Scholar
  19. 19.
    Khoshnevisan, D., Shi, Z.: Fast sets and points for fractional Brownian motion. Séminaire de Probabilités, XXXIV, pp. 393–416.Google Scholar
  20. 20.
    Khoshnevisan, D., Xiao, Y.: Lévy processes: capacity and Hausdorff dimension. Ann. Probab. 33(3), 841–878 (2005)Google Scholar
  21. 21.
    Khoshnevisan, D., Xiao, Y.: Weak unimodality of finite measures, and an application to potential theory of additive Lévy processes. Proc. Amer. Math. Soc. 131(8), 2611–2616 (electronic) (2003)Google Scholar
  22. 22.
    Khoshnevisan D. and Xiao Y. (2002). Level sets of additive Lévy processes. Ann. Probab. 30(1): 62–100 zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Khoshnevisan D., Xiao Y. and Zhong Y. (2003). Measuring the range of an additive Lévy process. Ann. Probab. 31(2): 1097–1141 zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Klenke A. and Mörters P. (2005). The multifractal spectrum of Brownian intersection local times. Ann. Probab. 33(4): 1255–1301 zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Lyons R. (1992). Random walks, capacity and percolation on trees. Ann. Probab. 20(4): 2043–2088 zbMATHMathSciNetGoogle Scholar
  26. 26.
    Lyons R. (1990). Random walks and percolation on trees. Ann. Probab. 18(3): 931–958 zbMATHMathSciNetGoogle Scholar
  27. 27.
    Mörters P. (2001). How fast are the particles of super-Brownian motion?. Probab. Theory Related Fields 121(2): 171–197 zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Peres, Y.: Remarks on intersection-equivalence and capacity-equivalence. Ann. Inst. H. Poincaré Phys. Théor. 64(3), 339–347 (1996a) (English, with English and French summaries)Google Scholar
  29. 29.
    Peres Y. (1996). Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys. 177(2): 417–434 zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Peres Y. and Steif J.E. (1998). The number of infinite clusters in dynamical percolation. Probab. Theory Related Fields 111(1): 141–165 zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Pruitt W.E. and Taylor S.J. (1969). Sample path properties of processes with stable components. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 12: 267–289 zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Taylor S.J. (1966). Multiple points for the sample paths of the symmetric stable process. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5: 247–264 zbMATHCrossRefMathSciNetGoogle Scholar

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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