Probability Theory and Related Fields

, Volume 140, Issue 1–2, pp 207–238 | Cite as

Volume growth and heat kernel estimates for the continuum random tree



In this article, we prove global and local (point-wise) volume and heat kernel bounds for the continuum random tree. We demonstrate that there are almost–surely logarithmic global fluctuations and log–logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r → 0. We also show that the on-diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t → 0 almost–surely. Finally, we prove that this quenched (almost–sure) behaviour contrasts with the local annealed (averaged over all realisations of the tree) volume and heat kernel behaviour, which is smooth.


Continuum random tree Brownian excursion Heat kernel estimates Volume fluctuations 

Mathematics Subject Classification (2000)

Primary: 60D05 Secondary: 60G57 Secondary: 60H25 Secondary: 60J35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aldous D. (1991). The continuum random tree I. Ann. Probab. 19(1): 1–28 MATHMathSciNetGoogle Scholar
  2. 2.
    Aldous, D.: The continuum random tree II. An overview. Stochastic analysis (Durham, 1990). Lond. Math. Soc. Lect. Note Ser., vol. 167. Cambridge University Press, Cambridge, pp. 23–70 (1991)Google Scholar
  3. 3.
    Aldous D. (1993). The continuum random tree III. Ann. Probab. 21(1): 248–289 MATHMathSciNetGoogle Scholar
  4. 4.
    Aldous D. (1993). Tree-based models for random distribution of mass. J. Stat. Phys. 73(3–4): 625–641 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Aldous D. (1994). Recursive self-similarity for random trees, random triangulations and Brownian excursion. Ann. Probab. 22(2): 527–545 MATHMathSciNetGoogle Scholar
  6. 6.
    Barlow, M.T.: Diffusions on fractals. Lectures on probability theory and statistics (Saint-Flour, 1995). Lecture Notes in Mathematics, vol. 1690. Springer, Berlin, pp. 1–121 (1998)Google Scholar
  7. 7.
    Barlow M.T., Bass R.F. and Kumagai T. (2006). Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Jpn. 58(2): 485–519 MATHMathSciNetGoogle Scholar
  8. 8.
    Barlow, M.T., Kumagai, T.: Random walk on the incipient infinite cluster on trees. Ill. J. Math. 50(1-4), 33–65 (2006, electronic)Google Scholar
  9. 9.
    Chung K.L. (1976). Excursions in Brownian motion. Ark. Mat. 14(2): 155–177 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ciesielski Z. and Taylor S.J. (1962). 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 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Croydon, D.A.: Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree. PreprintGoogle Scholar
  12. 12.
    Croydon, D.A.: Heat kernel fluctuations for a resistance form under non-uniform volume growth. In: Proceedings of the London Mathematical Society (to appear)Google Scholar
  13. 13.
    Duquesne, T., Le Gall, J. -F.: The Hausdorff measure of stable trees. PreprintGoogle Scholar
  14. 14.
    Duquesne, T., Le Gall, J.-F.: Random trees, Lévy processes and spatial branching processes. Astérisque 281:vi+147 (2002)Google Scholar
  15. 15.
    Duquesne T. and Le Gall J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Relat. Fields 131(4): 553–603 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Durrett R.T. and Iglehart D.L. (1977). Functionals of Brownian meander and Brownian excursion. Ann. Probab. 5(1): 130–135 MATHMathSciNetGoogle Scholar
  17. 17.
    Fukushima, M.: Dirichlet forms and Markov processes. North-Holland Mathematical Library, vol. 23. North-Holland, Amsterdam (1980)Google Scholar
  18. 18.
    Graf, S., Mauldin, R.D., Williams, S.C.: The exact Hausdorff dimension in random recursive constructions. Mem. Am. Math. Soc. 71(381): x + 121 (1988)Google Scholar
  19. 19.
    Hambly B.M. and Jones O.D. (2003). Thick and thin points for random recursive fractals. Adv. Appl. Probab. 35(1): 251–277 MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Hara T. and Slade G. (2000). The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion.. J. Math. Phys 41(3): 1244–1293 MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Kesten H. (1986). Sub-diffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Stat. 22(4): 425–487 MATHMathSciNetGoogle Scholar
  22. 22.
    Kigami J. (1995). Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128(1): 48–86 MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kigami, J.: Analysis on fractals. Cambridge Tracts in Mathematics, vol. 143. Cambridge University Press, Cambridge (2001)Google Scholar
  24. 24.
    Krebs W.B. (1995). Brownian motion on the continuum tree. Probab. Theory Relat. Fields 101(3): 421–433 MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Kumagai T. (2004). Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms. Publ. Res. Inst. Math. Sci. 40(3): 793–818 MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Marckert, J.F., Mokkadem, A.: Limit of normalized quadrangulations: the Brownian map. Ann. Prob. 34(6) (2006)Google Scholar
  27. 27.
    Revuz, D., Yor, M.: Continuous martingales and Brownian motion. 3rd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293. Springer, Berlin (1999)Google Scholar
  28. 28.
    Rogers, L.C.G., Williams, D.: Diffusions, Markov processes, and martingales, vol. 2. Cambridge Mathematical Library, Cambridge University Press, Cambridge, Ito calculus, Reprint of the second (1994) edition (2000)Google Scholar
  29. 29.
    Taylor S.J. and Tricot C. (1985). Packing measure and its evaluation for a Brownian path. Trans. Am. Math. Soc. 288(2): 679–699MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of WarwickCoventryUK

Personalised recommendations