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Packing and Hausdorff Measures of Stable Trees

Part of the Lecture Notes in Mathematics book series (LEVY,volume 2001)

Abstract

In this paper we discuss Hausdorff and packing measures of random continuous trees called stable trees. Stable trees form a specific class of Lévy trees (introduced by Le Gall and Le Jan in [33]) that contains Aldous’s continuum random tree which corresponds to the Brownian case. We provide results for the whole stable trees and for their level sets that are the sets of points situated at a given distance from the root. We first show that there is no exact packing measure for level sets. We also prove that non-Brownian stable trees and their level sets have no exact Hausdorff measure with regularly varying gauge function, which continues previous results from [14].

Keywords

  • Hausdorff measure
  • Lévy trees
  • Local time measure
  • Mass measure
  • Packing measure
  • Stable trees

AMS Subject Classification 2000

Primary: 60G57, 60J80

Secondary: 28A78

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References

  1. Abraham, R., Delmas, J.-F.: Fragmentation associated to Lévy processes using snake. Probab. Theor. Relat. Field. 141, 113–154 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Aldous, D.J.: The continuum random tree I. Ann. Probab. 19, 1–28 (1991)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Aldous, D.J.: The continuum random tree III. Ann. Probab. 21, 248–289 (1993)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Berlinkov, A.: Exact packing dimension in random recursive constructions. Probab. Theor. Relat. Field. 126, 477–496 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  6. Bingham, N.H.: Continuous branching processes and spectral positivity. Stoch. Process. Appl. 4, 217–242 (1976)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Bingham, N.H., Doney, R.A.: Asymptotic properties of super-critical branching processes. I: The Galton-Watson process. Annals of Applied Probab. 6, 711–731 (1974)

    MathSciNet  MATH  Google Scholar 

  8. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopaedia of Mathematics and its Applications, vol. 27. Cambridge University Press, Cambridge (1987)

    Google Scholar 

  9. Dress, A., Moulton, V., Terhalle, W.: T-theory: An overview. Eur. J. Combin. 17, 161–175 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Duquesne, T.: A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab. 31(2), 996–1027 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Duquesne, T.: The exact packing measure of Lévy trees. Preprint available on ArXiv (2010)

    Google Scholar 

  12. Duquesne, T., Le Gall, J.-F.: Random Trees, Lévy Processes and Spatial Branching Processes. Astérisque 281 (2002)

    Google Scholar 

  13. Duquesne, T., Le Gall, J.-F.: Probabilistic and fractal aspects of Lévy trees. Probab. Theor. Rel. Field. 131(4), 553–603 (2005)

    CrossRef  MATH  Google Scholar 

  14. Duquesne, T., Le Gall, J.-F.: The Hausdorff measure of stable trees. Alea 1, 393–415 (2006)

    MATH  Google Scholar 

  15. Duquesne, T., Le Gall, J.-F.: On the re-rooting invariance property of Lévy trees. Elect. Comm. Probab. 14, 317–326 (2009)

    MATH  Google Scholar 

  16. Duquesne, T., Winkel, M.: Growth of Lévy trees. Prob. Theor. Relat. Field. 139(3–4), 313–371 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Edgar, G.: Centered densities and fractal measures. New York J. Math. 13, 33–87 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Evans, S.: Probability and Real Trees. Saint-Flour Lectures Notes XXXV. Springer, Berlin (2005)

    Google Scholar 

  19. Evans, S., Pitman, J., Winter, A.: Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theor. Relat. Field. 134, 81–126 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II, 2nd edn. Wiley, New York (1971)

    Google Scholar 

  21. Goldschmidt, C., Haas, B.: Behavior near extinction time in self-similar fragmentation I: the stable case. Ann. IHP, 46(2), pp. 338–368 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics. Birkhäuser, Boston (1999)

    MATH  Google Scholar 

  23. Haas, B., Miermont, G.: The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electr. J. Probab. 9, 57–97 (2004)

    MathSciNet  Google Scholar 

  24. Haase, H.: The packing theorem and packing measure. Math. Nachr. 146, 77–84 (1990)

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. Helland, I.S.: Continuity of a class of random time transformations. Stoch. Process. Appl. 7, 79–99 (1978)

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. Jirina, M.: Stochastic branching processes with continous state-space. Czech. Math. J. 8, 292–313 (1958)

    MathSciNet  Google Scholar 

  27. Joyce, H.: Concerning the problem of subsets of finite positive packing measure. J. Lond. Math. Soc. 56, 557–566 (1997)

    CrossRef  MathSciNet  Google Scholar 

  28. Joyce, H.: A space on which diameter-type packing measure is not Borel regular. Proc. Amer. Math. Soc. 127, 985–991 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. Joyce, H., Preiss, D.: On the existence of subsets of positive finite packing measure. Mathematika 42, 14–24 (1995)

    CrossRef  MathSciNet  Google Scholar 

  30. Lamperti, J.: Continuous-state branching processes. Bull. Amer. Math. Soc. 73, 382–386 (1967)

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. Larman, D.G.: A new theory of dimension. Proc. Lond. Math. Soc. 17, 178–192 (1967)

    CrossRef  MathSciNet  MATH  Google Scholar 

  32. Le Gall, J.-F.: Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics. ETH, Zürich (1999)

    MATH  Google Scholar 

  33. Le Gall, J.-F., Le Jan, Y.: Branching processes in Lévy processes: The exploration process. Ann. Probab. 26(1), 213–252 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. Le Gall, J.-F., Perkins, E., Taylor, S.: The packing measure of the support of super-Brownian motion. Stoch. Process. Appl. 59, 1–20 (1995)

    CrossRef  MATH  Google Scholar 

  35. Miermont, G.: Self-similar fragmentations derived from the stable tree I: Splitting at heights. Probab. Theor. Relat. Field. 127(3), 423–454 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  36. Miermont, G.: Self-similar fragmentations derived from the stable tree II: Splitting at nodes. Probab. Theor. Relat. Field. 131(3), 341–375 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. Rogers, C., Taylor, S.: Functions continuous and singular with respect to Hausdorff measures. Mathematika 8, 1–31 (1961)

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. Silverstein, M.: A new approach to local times. J. Math. Mech. 17, 1023–1054 (1968)

    MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  40. Weill, M.: Regenerative real trees. Ann. Probab. 35(6), 2091–2121 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

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Duquesne, T. (2010). Packing and Hausdorff Measures of Stable Trees. In: Barndorff-Nielsen, O., Bertoin, J., Jacod, J., Klüppelberg, C. (eds) Lévy Matters I. Lecture Notes in Mathematics(), vol 2001. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14007-5_2

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