Hausdorff measure of arcs and Brownian motion on brownian spatial trees

Annals of Probability

Volumn 37, Issue 3, 2009, Pages 946-978

  Croydon, David A ^a  

^a UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

Branching random walk; Dawson Watanabe super process; Diffusion; Hausdorff measure; Random environment; Random walk; Scaling limit; Spatial tree

Indexed keywords

EID: 70349346417     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/08-AOP425     Document Type: Article

References (28)

1. ALDOUS, D. (1991). The continuum random tree. I. Ann. Probab. 19 1-28.
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.
3. ALDOUS, D. (1993). The continuum random tree. III. Ann. Probab. 21 248-289.
4. ALDOUS, D. (1993). Tree-based models for random distribution of mass. J. Statist. Phys. 73 625-641.
5. BILLINGSLEY, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York.
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.
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.
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.
9. CROYDON, D. A. (2008). Volume growth and heat kernel estimates for the continuum random tree. Probab. Theory Related Fields 140 207-238.
10. CROYDON, D. and HAMBLY, B. (2008). Self-similarity and spectral asymptotics for the continuum random tree. Stochastic Process. Appl. 118 730-754.
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.
12. DAWSON, D. A. and PERKINS, E. A. (1991). Historical Processes. Memoirs of the American Mathematical Society 93. Amer. Math. Soc., Providence, RI.
13. DUQUESNE, T. and LE GALL, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 1-147.
14. DUQUESNE, T. and LE GALL, J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 553-603.
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).
16. EVANS, S. N., PITMAN, J. andWINTER, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 81-126.
17. JANSON, S. andMARCKERT, J.-F. (2005). Convergence of discrete snakes. J. Theoret. Probab. 18 615-647.
18. KIGAMI, J. (1995). Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128 48-86.
19. KIGAMI, J. (2001). Analysis on Fractals. Cambridge Tracts in Mathematics 143. Cambridge Univ. Press, Cambridge.
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.
21. LE GALL, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel.
22. LE GALL, J.-F. (2006). Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15 35-62.
23. LE GALL, J.-F. and PERKINS, E. A. (1995). The Hausdorff measure of the support of twodimensional super-Brownian motion. Ann. Probab. 23 1719-1747.
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).
25. MATTILA, P. andMAULDIN, R. D. (1997). Measure and dimension functions: Measurability and densities. Math. Proc. Cambridge Philos. Soc. 121 81-100.
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.
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.