Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Hausdorff dimension of the contours of symmetric additive Lévy processes
Download PDF
Download PDF
  • Published: 27 February 2007

Hausdorff dimension of the contours of symmetric additive Lévy processes

  • Davar Khoshnevisan1,
  • Narn-Rueih Shieh2 &
  • Yimin Xiao3 

Probability Theory and Related Fields volume 140, pages 129–167 (2008)Cite this article

  • 95 Accesses

  • 4 Citations

  • Metrics details

An Erratum to this article was published on 08 November 2008

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.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Ayache A. and Xiao Y. (2005). Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets. J. Fourier Anal. Appl. 11: 407–439

    Article  MATH  MathSciNet  Google Scholar 

  2. Barlow, M.T., Perkins, E.: Levels at which every Brownian excursion is exceptional, Seminar on probability, XVIII, pp. 1–28 (1984)

  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

    Article  MATH  MathSciNet  Google Scholar 

  4. Bertoin J. (1996). Lévy Processes. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  5. Blath J. and Mörters P. (2005). Thick points of super-Brownian motion. Probab. Theory Related Fields 131(4): 604–630

    Article  MATH  MathSciNet  Google Scholar 

  6. Blumenthal R.M. and Getoor R.K. (1968). Markov Processes and Potential Theory. Academic Press, New York

    MATH  Google Scholar 

  7. Bochner S. (1955). Harmonic Analysis and the Theory of Probability. University of California Press, Berkeley and Los Angeles

    MATH  Google Scholar 

  8. Dalang R.C. and Nualart E. (2004). Potential theory for hyperbolic SPDEs. Ann. Probab. 32(3): 2099–2148

    Article  MATH  MathSciNet  Google Scholar 

  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. Dembo, A., Peres, Y., Rosen, J., Zeitouni, O.: Thick points for transient symmetric stable processes. Electron. J. Probab. 4(10) , 13 pp (electronic) (1999)

  11. Hawkes J. (1974). Local times and zero sets for processes with infinitely divisible distributions. J.~Lond. Math. Soc. 8: 517–525

    Article  MATH  MathSciNet  Google Scholar 

  12. Horowitz J. (1968). The Hausdorff dimension of the sample path of a subordinator. Israel J.~Math. 6: 176–182

    MATH  MathSciNet  Google Scholar 

  13. Kahane J.-P. (1985). Some Random Series of Functions. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  14. Khoshnevisan, D.: The codimension of the zeros of a stable process in random scenery, Séminaire de Probabilités XXXVII, pp. 236–245 (2003)

  15. Khoshnevisan D. (2002). Multiparameter Processes. Springer, New York

    MATH  Google Scholar 

  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)

  17. Khoshnevisan D., Levin D.A. and Méndez-Hernández P.J. (2005). On dynamical Gaussian random walks. Ann. Probab. 33(4): 1452–1478

    Article  MATH  MathSciNet  Google Scholar 

  18. Khoshnevisan, D., Peres, Y., Xiao, Y.: Limsup random fractals. Electron. J. Probab. 5(5), 24 pp. (electronic) (2000)

  19. Khoshnevisan, D., Shi, Z.: Fast sets and points for fractional Brownian motion. Séminaire de Probabilités, XXXIV, pp. 393–416.

  20. Khoshnevisan, D., Xiao, Y.: Lévy processes: capacity and Hausdorff dimension. Ann. Probab. 33(3), 841–878 (2005)

    Google Scholar 

  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. Khoshnevisan D. and Xiao Y. (2002). Level sets of additive Lévy processes. Ann. Probab. 30(1): 62–100

    Article  MATH  MathSciNet  Google Scholar 

  23. Khoshnevisan D., Xiao Y. and Zhong Y. (2003). Measuring the range of an additive Lévy process. Ann. Probab. 31(2): 1097–1141

    Article  MATH  MathSciNet  Google Scholar 

  24. Klenke A. and Mörters P. (2005). The multifractal spectrum of Brownian intersection local times. Ann. Probab. 33(4): 1255–1301

    Article  MATH  MathSciNet  Google Scholar 

  25. Lyons R. (1992). Random walks, capacity and percolation on trees. Ann. Probab. 20(4): 2043–2088

    MATH  MathSciNet  Google Scholar 

  26. Lyons R. (1990). Random walks and percolation on trees. Ann. Probab. 18(3): 931–958

    MATH  MathSciNet  Google Scholar 

  27. Mörters P. (2001). How fast are the particles of super-Brownian motion?. Probab. Theory Related Fields 121(2): 171–197

    Article  MATH  MathSciNet  Google Scholar 

  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)

  29. Peres Y. (1996). Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys. 177(2): 417–434

    Article  MATH  MathSciNet  Google Scholar 

  30. Peres Y. and Steif J.E. (1998). The number of infinite clusters in dynamical percolation. Probab. Theory Related Fields 111(1): 141–165

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  32. Taylor S.J. (1966). Multiple points for the sample paths of the symmetric stable process. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5: 247–264

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Utah, 155 S. 1400 E., Salt Lake City, UT, 84112–0090, USA

    Davar Khoshnevisan

  2. Department of Mathematics, National Taiwan University, Taipei, 10617, Taiwan

    Narn-Rueih Shieh

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

    Yimin Xiao

Authors
  1. Davar Khoshnevisan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Narn-Rueih Shieh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yimin Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Davar Khoshnevisan.

Additional information

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.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

  • Received: 09 March 2006

  • Revised: 30 January 2007

  • Published: 27 February 2007

  • Issue Date: January 2008

  • DOI: https://doi.org/10.1007/s00440-007-0060-7

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Additive Lévy processes
  • Level sets
  • Hausdorff dimension

Mathematics Subject Classification (2000)

  • 60G70
  • 60F15
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature