Local Times and Related Properties of Multidimensional Iterated Brownian Motion

Journal of Theoretical Probability

Volumn 11, Issue 2, 1998, Pages 383-408

  Xiao, Yimin ^a  

^a UNIVERSITY OF UTAH   (United States)

Author keywords

Hausdorff dimension; H lder conditions; Iterated Brownian motion; Level set; Local times

Indexed keywords

EID: 0032359434     PISSN: 08949840     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022679721638     Document Type: Article

References (35)

1. Adler, R. J. (1978). The uniform dimension of the level sets of a Brownian sheet. Ann. Prob. 6, 509-515.
2. Adler, R. J. (1978). The Geometry of Random Fields, John Wiley, New York.
3. Berman, S. M. (1981). Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc. 137, 277-299.
4. Berman, S. M. (1972). Gaussian sample function: uniform dimension and Hölder conditions nowhere. Nagoya Math. J. 46, 63-86.
5. Berman, S. M. (1973). Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23, 69-94.
6. Bertoin, J. (1996). Iterated Brownian motion and stable (1/4) subordinator. Stat. Prob. Lett. 27, 111-114.
7. Burdzy, K. (1993). Some path properties of iterated Brownian motion. In Chung, K. L., Cinlar, E., and Sharp, M. J., (eds.), Seminar on Stochastic Processes 1992, Birkhäuser, Boston, pp. 67-87.
8. Burdzy, K. (1994). Variation of iterated Brownian motion. In Dawson, D. A., (ed.), Measure-Valued Processes, Stochastic Partial Differential Equations and Interacting Systems, RM Proceedings and Lecture Notes 5, 35-53.
9. Burdzy, K., and Khoshnevisan, D. (1995). The level sets of iterated Brownian motion. Séminaire de Probabilité XXIX, Lecture Notes in Math. 1613, 231-236.
10. Csáki, E., Csörgo, M., Földes, A., and Révész, P. (1995). Global Strassen-type theorems for iterated Brownian motion. Stoch. Proc. Appl. 59, 321-341.
11. Csáki, E., Csörgo, M., Földes, A., and Révész, P. (1996). The local time of iterated Brownian motion. J. Theoret. Prob. 9, 745-763.
12. Deheuvels, P., and Mason, D. M. (1992). A functional LIL approach to pointwise Bahadur-Kiefer theorems. In Dudley, R. M., Hahn, M. G., and Kuelbs, J., (eds.), Prob, in Banach Spaces 8, 255-266.
13. Ehm, W. (1981). Sample function properties of multi-parameter stable processes. Z. Wahrsch. verw. Geb. 56, 195-228.
14. Falconer, K. J. (1990). Fractal Geometry - Mathematical Foundations and Applications, John Wiley and Sons.
15. Garsia, A. (1970). Continuity properties for multidimensional Gaussian processes. Proc. of the Sixth Berkeley Symp. on Math. Stat. and Prob. 2, 369-374.
16. Geman, D., and Horowitz, J. (1980). Occupation densities. Ann. Prob. 8, 1-67.
17. Geman, D., Horowitz, J., and Rosen, J. (1984). A local time analysis of intersections of Brownian paths in the plane. Ann. Prob. 12, 86-107.
18. Hu, Y., Pierre-Lotti-Viaud, D., and Shi, Z. (1995). Laws of the iterated logarithm for iterated Brownian motion. J. Theoret. Prob. 8, 303-319.
19. Hu, Y., and Shi, Z. (1995). The Csörgo-Révész modulus of non-differentiability of iterated Brownian motion. Stoch. Proc. Appl. 58, 167-179.
20. Kahane, J.-P. (1985). Some Random Series of Functions, Second Edition, Cambridge University Press.
21. Kaufman, R. (1969). Une propriété métrique du mouvement brownien. C. R. Acad. Sci. Paris 268, 727-728.
22. Kaufman, R. (1985). Temps locaux et dimensions. C. R. Acad. Sci. Paris 300, 281-282.
23. Kôno, N. (1977). Hölder conditions for the local times of certain Gaussian processes with stationary increments. Proc. Japan Acad. Series A 53, 84-87.
24. Kôno, N. (1991). Recent development on random fields and their sample paths, part III: Self-similar processes. Soochow J. Math. 17, 327-361.
25. Khoshnevisan, D., and Lewis, T. M. (1996a). The modulus of continuity for iterated Brownian motion. J. Theoret. Prob. 9, 317-333.
26. Khoshnevisan, D., and Lewis, T. M. (1996b). Chung's law of the iterated logarithm for iterated Brownian motion. Ann. Inst. Henri Poincaré: Prob. et Stat. 32, 349-359.
27. Monrad, D., and Pitt, L. D. (1987). Local nondeterminism and Hausdorff dimension. In Cinlar, E., Chung, K. L., and Getoor, R. K., (eds.), Prog. in Probab. and Statist., Seminar on Stochastic Processes 1986, Birkhäuser, pp. 163-189.
28. Perkins, E. (1981). The exact Hausdorff measure of the level sets of Brownian motion. Z. Warsch. verw. Geb. 58, 373-388.
29. Perkins, E., and Taylor, S. J. (1987). Uniform measure results for the image of subsets under Brownian motion. Prob. Th. Rel. Fields 76, 257-289.
30. Pitt, L. D. (1978). Local times for Gaussian vector fields. Indiana Univ. Math. J. 27, 309-330.
31. Rogers, C. A., and Taylor, S. J. (1961). Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8, 1-31.
32. Shi, Z. (1995). Lower limits of iterated Wiener processes. Stat. Prob. Lett. 23, 259-270.
33. Taqqu, M. S. (1986). A bibliographical guide to self-similar processes and long-range dependence. Progress in Prob. Stat. 11, 137-162. Dependence in Prob. and Stat. (eds.), Eberlein, M. S, Taqqu.
34. Taylor, S. J., and Tricot, C. (1985). Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288, 679-699.
35. Xiao, Yimin (1997). Hölder conditions for local times of Gaussian random fields. Probab. Theory Related Fields 109, 129-157.