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
Uniform measure results for the image of subsets under Brownian motion
Download PDF
Download PDF
  • Published: September 1987

Uniform measure results for the image of subsets under Brownian motion

  • Edwin A. Perkins1 &
  • S. James Taylor2 

Probability Theory and Related Fields volume 76, pages 257–289 (1987)Cite this article

  • 128 Accesses

  • 35 Citations

  • Metrics details

Summary

The paper obtains bounds on the Hausdorff and packing measures of the imageX(E) of a Borel setE by a transient strictly stable processX t which a.s. hold for allE and for every measure function\(h_{\beta ,\gamma } (s) = s^\beta \left| {\log s} \right|^{\gamma ^ \star }\). In some cases examples are constructed to show that the bounds are sharp.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Blumenthal, R.M., Getoor, R.K.: A dimension theorem for sample functions of stable processes. Ill. J. Math.4, 370–375 (1960)

    Google Scholar 

  2. Blumenthal, R.M., Getoor, R.K., Ray, D.B.: On the distribution of first hits for the symmetric stable processes. Trans. Am. Math. Soc.99, 540–554 (1961)

    Google Scholar 

  3. Ciesielski, Z., Taylor, S.J.: First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Am. Math. Soc.103, 434–450 (1962)

    Google Scholar 

  4. Erdös, P., Taylor, S.J.: On the Hausdorff measure of Brownian paths in the plane. Proc. Cambr. Phil. Soc.57, 209–222 (1961)

    Google Scholar 

  5. Hawkes, J.: A lower Lipshitz condition for the stable subordinator. Z. Wahrscheinlichkeitstheor. Verw. Geb.17, 23–32 (1971)

    Google Scholar 

  6. Hawkes, J., Pruitt, W.E.: Uniform dimension results for processes with independent increments. Z. Wahrscheinlichkeitstheor. Verw. Geb.28, 277–288 (1974)

    Google Scholar 

  7. Hendricks, W.J.: Hausdorff dimension in a process with stable components—an interesting counter-example. Ann. Math. Statist.43, 690–694 (1972)

    Google Scholar 

  8. Kaufman, R.: Une propriété metriqué du mouvement brownien. C. R. Acad. Sci., Paris268, 727–728 (1969)

    Google Scholar 

  9. Le Gall, J.-F.: Le comportement du mouvement brownien entre les deux instants où il passe par un point double. J. Funct. Anal.71, 246–262 (1987)

    Google Scholar 

  10. McKean, H.P.: Hausdorff-Besicovitch dimension of Brownian motion paths. Duke Math. J.22, 229–234 (1955)

    Google Scholar 

  11. Orey, S., Taylor, S.J.: How often on a Brownian path does the law of iterated logarithm fail? Proc. Lond. Math. Soc.28, 174–192 (1974)

    Google Scholar 

  12. Perkins, E.A., Taylor, S.J.: Measuring close approaches on a Brownian path. Ann. Probab. (to appear)

  13. Pruitt, W.E., Taylor, S.J.: The potential kernel and hitting probabilities for the general stable process in ∝N. Trans. Am. Math. Soc.146, 299–321 (1969)

    Google Scholar 

  14. Spitzer, F.: Some theorems concerning two-dimensional Brownian motion. Trans. Am. Math. Soc.87, 187–197 (1958)

    Google Scholar 

  15. Takeuchi, J.: On the sample paths of the symmetric stable processes in space. J. Math. Soc. Japan16, 109–127 (1964)

    Google Scholar 

  16. Taylor, S.J.: Sample path properties of a transient stable process. J. Math. Mech.16, 1229–1246 (1967)

    Google Scholar 

  17. Taylor, S.J.: Regularity of irregularities on a Brownian path. Ann. Inst. Fourier, Grenoble24, 195–203 (1964)

    Google Scholar 

  18. Taylor, S.J.: The use of packing measure in the analysis of random sets. Proc. 15th Symposium on Stochastic Processes and Applications. Berlin Heidelberg New York: Springer (1986)

    Google Scholar 

  19. Taylor, S.J.: The measure theory of random fractals. Math. Proc. Cambr. Phil. Soc.100, 383–406 (1986)

    Google Scholar 

  20. Taylor, S.J., Tricot, C.: Packing measure and its evaluation for a Brownian path. Trans. Am. Math. Soc.288, 679–699 (1985)

    Google Scholar 

  21. Hawkes, J.: Image and intersection sets for subordinators. J. London Math. Soc. (2)17, 567–576 (1978)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, V6T 1Y4, Vancouver, BC, Canada

    Edwin A. Perkins

  2. Department of Mathematics, University of Virginia, 22903, Charlottesville, VA, USA

    S. James Taylor

Authors
  1. Edwin A. Perkins
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. James Taylor
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

While preparing this paper, the author was partially supported by NSERC and by NSF on contract #DMS-8317815

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Perkins, E.A., Taylor, S.J. Uniform measure results for the image of subsets under Brownian motion. Probab. Th. Rel. Fields 76, 257–289 (1987). https://doi.org/10.1007/BF01297485

Download citation

  • Received: 19 December 1986

  • Issue Date: September 1987

  • DOI: https://doi.org/10.1007/BF01297485

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

  • Stochastic Process
  • Brownian Motion
  • Probability Theory
  • Measure Function
  • Mathematical Biology
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