The Annals of Probability

The packing measure of the range of Super-Brownian motion

Thomas Duquesne

Full-text: Open access


We prove that the total range of Super-Brownian motion with quadratic branching mechanism has an exact packing measure with respect to the gauge function g(r)=r4(log log 1/r)−3 in super-critical dimensions d≥5. More precisely, we prove that the total occupation measure of Super-Brownian motion is equal to the g-packing measure restricted to its range, up to a deterministic multiplicative constant that only depends on space dimension d.

Article information

Ann. Probab., Volume 37, Number 6 (2009), 2431-2458.

First available in Project Euclid: 16 November 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G57: Random measures 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 28A78: Hausdorff and packing measures

Super-Brownian motion Brownian Snake range exact packing measure


Duquesne, Thomas. The packing measure of the range of Super-Brownian motion. Ann. Probab. 37 (2009), no. 6, 2431--2458. doi:10.1214/09-AOP468.

Export citation


  • [1] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
  • [2] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [3] Besicovitch, A. S. (1945). A general form of the covering principle and relative differentiation of additive functions. Proc. Cambridge Philos. Soc. 41 103–110.
  • [4] Bismut, J.-M. (1985). Last exit decompositions and regularity at the boundary of transition probabilities. Z. Wahrsch. Verw. Gebiete 69 65–98.
  • [5] Ciesielski, Z. and Taylor, J. (1962). First passage and sojourn time and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103 434–452.
  • [6] Dawson, D. A., Iscoe, I. and Perkins, E. A. (1989). Super-Brownian motion: Path properties and hitting probabilities. Probab. Theory Related Fields 83 135–205.
  • [7] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 vi–147.
  • [8] Duquesne, T. and Le Gall, J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 553–603.
  • [9] Dynkin, E. B. (1994). An Introduction to Branching Measure-Valued Processes. CRM Monograph Series 6. Amer. Math. Soc., Providence, RI.
  • [10] Edgar, G. A. (2007). Centered densities and fractal measures. New York J. Math. 13 33–87 (electronic).
  • [11] Jain, N. C. and Pruitt, W. E. (1987). Lower tail probability estimates for subordinators and nondecreasing random walks. Ann. Probab. 15 75–101.
  • [12] Le Gall, J.-F. (1991). Brownian excursions, trees and measure-valued branching processes. Ann. Probab. 19 1399–1439.
  • [13] Le Gall, J.-F. (1993). A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Related Fields 95 25–46.
  • [14] Le Gall, J.-F. (1993). The uniform random tree in a Brownian excursion. Probab. Theory Related Fields 96 369–383.
  • [15] Le Gall, J.-F. (1994). Hitting probilities and potential theory for the Brownian path-valued process. Ann. Inst. Fourier 44 237–251.
  • [16] Le Gall, J.-F. (1994). A path-valued Markov process and its connection with partial differential equations. In Proc. First European Congress of Mathematics, Vol. II (A. Joseph, F. Mignot, F. Murat, B. Prum and R. Rentschler, eds.) 185–212. Birkhaüser, Boston.
  • [17] Le Gall, J.-F. (1999). The Hausdorff measure of the range of super-Brownian motion. In Perplexing Problems in Probability, Festschrift in Honor of Harry Kesten (M. Bramson and R. Durret, eds.) 285–314. Birkhäuser, Boston.
  • [18] Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.
  • [19] Le Gall, J.-F., Perkins, E. A. and Taylor, S. J. (1995). The packing measure of the support of super-Brownian motion. Stochastic Process. Appl. 59 1–20.
  • [20] Perkins, E. (2002). Dawson–Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics (Saint-Flour, 1999). Lecture Notes in Math. 1781 125–324. Springer, Berlin.
  • [21] Pitman, J. W. (1975). One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. in Appl. Probab. 7 511–526.
  • [22] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin.
  • [23] Skorohod, A. V. (1961). Asymptotic formulas for stable distribution laws. In Select. Transl. Math. Statist. and Probability 1 157–161. IMS and Amer. Math. Soc., Providence, RI.
  • [24] Slade, G. (1999). Lattice trees, percolation and super-Brownian motion. In Perplexing Problems in Probability. Progress in Probability 44 35–51. Birkhäuser, Boston, MA.
  • [25] Taylor, S. J. and Tricot, C. (1985). Packing measure, and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288 679–699.