Local times of fractional Brownian sheets

Probability Theory and Related Fields

Volumn 124, Issue 2, 2002, Pages 204-226

  Xiao, Yimin ^a   Zhang, Tusheng ^b  

^a MICHIGAN STATE UNIVERSITY   (United States)

^b UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Fractional Brownian motion; Fractional Brownian sheet; Joint continuity; Local times

Indexed keywords

EID: 0035981340     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s004400200210     Document Type: Article

References (28)

1. Adler, R.J.: The uniform dimension of the level sets of a Brownian sheet. Ann. of Probab. 6, 509-515 (1978)
2. Adler, R.J.: The Geometry of Random Fields. Wiley, New York, 1981
3. Bapat, R.B., Raghavan, T.E.S.: Non-negative Matrices and Applications. Cambridge University Press, 1996
4. Berman, S.M.: Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc. 137, 277-299 (1969)
5. Berman, S.M.: Gaussian sample function: uniform dimension and Hölder conditions nowhere. Nagoya Math. J. 46, 63-86 (1972)
6. Berman, S.M.: Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23, 69-94 (1973)
7. Cuzick, J.: Continuity of Gaussian local times. Ann. of Probab. 10, 818-823 (1982)
8. Cuzick, J., Du Peez, J.: Joint continuity of Gaussian local times. Ann. of Probab. 10, 810-817 (1982)
9. Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Fractional Brownian motion and stochastic equations in Hilbert spaces. Preprint, 2000
10. Dunker, T.: Estimates for the small ball probabilities of the fractional Brownian sheet. J. Theoret. Probab. 13, 357-382 (2000)
11. Ehm, W.: Sample function properties of multi-parameter stable processes. Z. Wahrsch. verw Gebiete 56, 195-228 (1981)
12. Eisenbaum, N., Khoshnevisan, D.: On the most visited sites of symmetric Markov processes. Preprint, (2001)
13. Garsia, A.M.: Continuity properties of Gaussian processes with multidimensional time parameter. Proc. 6th Berkeley Symp. Math. Statist. Probability II, pp. 369-374, (1971)
14. Geman, D., Horowitz, J.: Occupation densities. Ann. of Probab. 8, 1-67 (1980)
15. Geman, D., Horowitz, J., Rosen, J.: A local time analysis of intersections of Brownian paths in the plane. Ann. of Probab. 12, 86-107 (1984)
16. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, 1999
17. Hu, Y., Øksendal, B., Zhang, T.: Stochastic partial differential equations driven by multiparameter fractional white noise. Stochastic Processes, Physics and Geometry: new interplays, II (Leipzig, 1999), 327-337, Amer. Math. Soc., Providence, RI, 2000
18. Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422-437 (1968)
19. Mason, D.M., Shi, Z.: Small deviations for some multi-parameter Gaussian processes. J. Theoret. Probab. 14, 213-239 (2001)
20. Monrad, D., Pitt, L.D.: Local nondeterminism and Hausdorff dimension. Prog. in Probab. and Statist., Seminar on Stochastic Processes (1986), (Cinlar, E., Chung, K.L., Getoor, R.K. Editors), pp 163-189, Birkhauser, 1987
21. Øksendal, B., Zhang, T.: Multiparameter fractional Brownian motion and quasi-linear stochastic partial differential equations. Stochastics and Stochastics Reports, 71, 141-163 (2000)
22. Orey, S.: Gaussian sample functions and the Hausdorff dimension of level crossing. Z. Wahrsch. verw Gebiete 15, 249-256 (1970)
23. Orey, S., Pruitt, W.E.: Sample functions of the N-parameter Wiener process. Ann. of Probab. 1, 138-163 (1973)
24. Pitt, L.D.: Local times for Gaussian vector fields. Indiana Univ. Math. J. 27, 309-330 (1978)
25. Rosen, J.: Self-intersections of random fields. Ann. of Probab. 12, 108-119 (1984)
26. Samorodnitsky, G., Taqqu, M.S.: Stable non-Gaussian Random Processes: Stochastic models with infinite variance. Chapman & Hall, New York, 1994
27. Talagrand, M.: Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann. of Probab. 23, 767-775 (1995)
28. Xiao, Y.: Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Th. Rel. Fields 109, 129-157 (1997)