Hausdorff measure of arcs and Brownian motion on Brownian spatial trees. (English) Zbl 1219.60052

Summary: A Brownian spatial tree is defined to be a pair \(({\mathcal T},\varphi)\), where \(\mathcal T\) is the rooted real tree naturally associated with a Brownian excursion and \(\varphi \) is a random continuous function from \(\mathcal T\) into \(\mathbb R^d\) such that, conditional on \({\mathcal T}, \varphi \) maps each arc of \(\mathcal T\) to the image of a Brownian motion path in \(\mathbb R^d\) run for a time equal to the arc length. It is shown that, in high dimensions, the Hausdorff measure of arcs can be used to define an intrinsic metric \(d_{\mathcal S}\) on the set \({\mathcal S}: = \varphi ({\mathcal T})\). Applications of this result include the recovery of the spatial tree \(({\mathcal T}, \varphi)\) from the set \(\mathcal S\) alone, which implies in turn that a Dawson-Watanabe super-process can be recovered from its range. Furthermore, \(d_{\mathcal S}\) can be used to construct a Brownian motion on \({\mathcal S}\), which is proved to be the scaling limit of simple random walks on related discrete structures. In particular, a limiting result for the simple random walk on the branching random walk is obtained.


60G57 Random measures
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments
28A78 Hausdorff and packing measures
Full Text: DOI arXiv


[1] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1-28. · Zbl 0722.60013
[2] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis ( Durham , 1990). London Mathematical Society Lecture Note Series 167 23-70. Cambridge Univ. Press, Cambridge. · Zbl 0791.60008
[3] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248-289. · Zbl 0791.60009
[4] Aldous, D. (1993). Tree-based models for random distribution of mass. J. Statist. Phys. 73 625-641. · Zbl 1102.60318
[5] Billingsley, P. (1999). Convergence of Probability Measures , 2nd ed. Wiley Series in Probability and Statistics : Probability and Statistics . Wiley, New York. · Zbl 0944.60003
[6] Burdzy, K. (1993). Some path properties of iterated Brownian motion. In Seminar on Stochastic Processes , 1992 ( Seattle, WA , 1992). Progress in Probability 33 67-87. Birkhäuser, Boston, MA. · Zbl 0789.60060
[7] 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. Amer. Math. Soc. 103 434-450. · Zbl 0121.13003
[8] Croydon, D. A. (2008). Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree. Ann. Inst. H. Poincaré Probab. Statist. 44 987-1019. · Zbl 1187.60083
[9] Croydon, D. A. (2008). Volume growth and heat kernel estimates for the continuum random tree. Probab. Theory Related Fields 140 207-238. · Zbl 1133.62066
[10] Croydon, D. and Hambly, B. (2008). Self-similarity and spectral asymptotics for the continuum random tree. Stochastic Process. Appl. 118 730-754. · Zbl 1143.60012
[11] 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. · Zbl 0692.60063
[12] Dawson, D. A. and Perkins, E. A. (1991). Historical Processes. Memoirs of the American Mathematical Society 93 . Amer. Math. Soc., Providence, RI. · Zbl 0754.60062
[13] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 1-147. · Zbl 1037.60074
[14] Duquesne, T. and Le Gall, J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 553-603. · Zbl 1070.60076
[15] Duquesne, T. and Le Gall, J.-F. (2006). The Hausdorff measure of stable trees. ALEA Lat. Am. J. Probab. Math. Stat. 1 393-415 (electronic). · Zbl 1128.60072
[16] Evans, S. N., Pitman, J. and Winter, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 81-126. · Zbl 1086.60050
[17] Janson, S. and Marckert, J.-F. (2005). Convergence of discrete snakes. J. Theoret. Probab. 18 615-647. · Zbl 1084.60049
[18] Kigami, J. (1995). Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128 48-86. · Zbl 0820.60060
[19] Kigami, J. (2001). Analysis on Fractals. Cambridge Tracts in Mathematics 143 . Cambridge Univ. Press, Cambridge. · Zbl 0998.28004
[20] Le Gall, J.-F. (1999a). The Hausdorff measure of the range of super-Brownian motion. In Perplexing Problems in Probability. Progress in Probability 44 285-314. Birkhäuser, Boston, MA. · Zbl 0945.60039
[21] Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich . Birkhäuser, Basel. · Zbl 0938.60003
[22] Le Gall, J.-F. (2006). Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15 35-62. · Zbl 1129.60047
[23] Le Gall, J.-F. and Perkins, E. A. (1995). The Hausdorff measure of the support of two-dimensional super-Brownian motion. Ann. Probab. 23 1719-1747. · Zbl 0856.60055
[24] Marckert, J.-F. and Mokkadem, A. (2003). States spaces of the snake and its tour-convergence of the discrete snake. J. Theoret. Probab. 16 1015-1046 (2004). · Zbl 1044.60083
[25] Mattila, P. and Mauldin, R. D. (1997). Measure and dimension functions: Measurability and densities. Math. Proc. Cambridge Philos. Soc. 121 81-100. · Zbl 0885.28005
[26] 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. · Zbl 1020.60075
[27] Slade, G. (2006). The Lace Expansion and Its Applications. Lecture Notes in Math. 1879 . Springer, Berlin. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6-24, 2004, edited and with a foreword by Jean Picard.
[28] Tribe, R. (1994). A representation for super Brownian motion. Stochastic Process. Appl. 51 207-219. · Zbl 0810.60075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.