Hausdorff-type measures of the sample paths of fractional Brownian motion

Stochastic Processes and their Applications

Volumn 74, Issue 2, 1998, Pages 251-272

  Xiao, Yimin ^a  

^a UNIVERSITY OF UTAH   (United States)

Author keywords

60G15; 60G17; Fractional Brownian motion; Graph set; Hausdorff type measures; Image; Packing measure; Primary 28A78

Indexed keywords

EID: 0000060868     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(97)00119-1     Document Type: Article

References (28)

1. Ciesielski Z., Taylor S.J. First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103:1962;434-450.
2. Ehm W. Sample function properties of multi-parameter stable processes. Z. Wahrsch. verw Gebiete. 56:1981;195-228.
3. Falconer K.J. Fractal Geometry - Mathematical Foundations And Applications. 1990;Wiley, New York.
4. Goldman A., 1988. Mouvement Brownienà plusieurs paramètres: mesure de Hausdorff des trajectoires. Astérisque 167.
5. Kahane, J-P., 1985. Some Random Series of Functions, 2nd ed. Cambridge University Press, Cambridge.
6. Jain N.C., Pruitt S.E. The correct measure function for the graph of a transient stable process. Z. Wahrsch. verw. Gebiete. 9:1968;131-138.
7. Lévy P. La mesure de Hausdorff de la courbe du mouvement brownien. Giorn. Ist. Ital. Attuari. 16:1953;1-37.
8. Mattila P. Geometry of sets and measures in Euclidean spaces. 1995;Cambridge University Press, Cambridge.
9. Monrad D., Rootzén H. Small values of Gaussian processes and functional laws of the iterated logarithm. Probab. Theory Related Fields. 101:1995;173-192.
10. Orey S., Pruitt W.E. Sample functions of the. N -parameter Wiener process Ann. Probab. 1:1973;138-163.
11. Pitt L.D., Tran L.T. Local sample path properties of Gaussian fields. Ann. Probab. 7:1979;477-493.
12. Pruitt W.E., Taylor S.J. Sample path properties of processes with stable components. Z. Wahrsch. verw. Gebiete. 12:1969;267-289.
13. Ray D. Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion. Trans. Amer. Math. Soc. 106:1964;436-444.
14. Rogers C.A. Hausdorff Measures. 1970;Cambridge University Press, Cambridge.
15. Rogers C.A., Taylor S.J. Functions continuous and singular with respect to a Hausdorff measure. Mathematika. 8:1961;1-31.
16. Saint Raymond X, Tricot, C, Packing regularity of sets in n -space. Math. Proc. Phil. Soc. 103 (1988) 133-145.
17. Talagrand M. New Gaussian estimates for enlarged balls. Geom. Funct. Anal. 3:1993;502-520.
18. Talagrand M. Hausdorff measure of the trajectories of multiparameter fractional Brownian motion. Ann. Probab. 23:1995;767-775.
19. Talagrand M. Lower classes for fractional Brownian motion. J. Theoret. Probab. 9:1996;191-213.
20. Talagrand, M., 1996b. Multiple points of trajectories of multiparameter fractional Brownian motion. Preprint.
21. Talagrand M., Xiao Y. Fractional Brownian motion and packing dimension. J. Theoret. Probab. 9:1996;579-593.
22. Taylor S.J. The exact Hausdorff measure of the sample path for planar Brownian motion. Proc. Cambridge Phil. Soc. 60:1964;253-258.
23. Taylor S.J. The measure theory of random fractals. Math. Proc. Cambridge Philos. Soc. 100:1986;383-406.
24. Taylor S.J., Tricot C. Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288:1985;679-699.
25. Xiao Y. Hausdorff measure of the sample paths of Gaussian random fields. Osaka J. Math. 33:1996;895-913.
26. Xiao Y. Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Related Fields. 109:1997;129-157.
27. Xiao, Y., 1997b. Hausdorff measure of the graph of fractional Brownian motion. Math. Proc. Cambridge Philos. Soc. 122, 565-576.
28. Xiao, Y., 1997c. Fractal measures of the sets associated to Gaussian random fields. In: Trends in Probability and Related Analysis: Proc. Symp. on Analysis and Probability 1996, in press.