Advertisement

Recent Developments on Fractal Properties of Gaussian Random Fields

  • Yimin Xiao
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

We review some recent developments in studying fractal and analytic properties of Gaussian random fields. It is shown that various forms of strong local nondeterminism are useful for studying many fine properties of Gaussian random fields. A list of open questions is included.

Keywords

Brownian Motion Hausdorff Dimension Fractional Brownian Motion Gaussian Random Field Stationary Increment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The author thanks the referee for his/her helpful comments which have led to improvement of the manuscript. Research partially supported by the NSF grant DMS-1006903.

References

  1. 1.
    Adler, R.J.: The Geometry of Random Fields. Wiley, New York (1981)MATHGoogle Scholar
  2. 2.
    Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Springer, New York (2007)MATHGoogle Scholar
  3. 3.
    Ayache, A., Wu, D., Xiao, Y.: Joint continuity of the local times of fractional Brownian sheets. Ann. Inst. H. Poincaré Probab. Statist. 44, 727–748 (2008)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Ayache, A., Xiao, Y.: Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets. J. Fourier Anal. Appl. 11, 407–439 (2005)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Baraka, D., Mountford, T.S.: A law of iterated logarithm for fractional Brownian motions. In: Séminaire de Probabilités XLI. Lecture Notes in Mathematics, vol. 1934, pp. 161–179. Springer, Berlin (2008)Google Scholar
  6. 6.
    Baraka, D., Mountford, T.S.: The exact Hausdorff measure of the zero set of fractional Brownian motion. J. Theor. Probab. 24, 271–293 (2011)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Baraka, D., Mountford, T.S., Xiao, Y.: Hölder properties of local times for fractional Brownian motions. Metrika 69, 125–152 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Benassi, A., Jaffard, S., Roux, D.: Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13, 19–90 (1997)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Berman, S.M.: Gaussian sample function: uniform dimension and Hölder conditions nowhere. Nagoya Math. J. 46, 63–86 (1972)MathSciNetMATHGoogle Scholar
  10. 10.
    Berman, S.M.: Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23, 69–94 (1973)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Biermé, H., Lacaux, C., Xiao, Y.: Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields. Bull. London Math. Soc. 41, 253–273 (2009)MATHCrossRefGoogle Scholar
  12. 12.
    Biermé, H., Meerschaert, M.M., Scheffler, H.-P.: Operator scaling stable random fields. Stoch. Process. Appl. 117, 313–332 (2007)CrossRefGoogle Scholar
  13. 13.
    Blath, J., Martin, A.: Propagation of singularities in the semi-fractional Brownian sheet. Stoch. Process. Appl. 118, 1264–1277 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Bonami, A., Estrade, A.: Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9, 215–236 (2003)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Chen, X., Li, W.V., Rosinski, J., Shao, Q.-M.: Large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes. Ann. Probab. 39, 729–778 (2011)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Chen, Z., Xiao, Y.: On intersections of independent anisotropic Gaussian random fields. Sci. China Math. 55, 2217–2232 (2012)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Christakos, M.J.: Modern Spatiotemporal Geostatistics. Oxford University Press, Oxford (2000)Google Scholar
  18. 18.
    Cuzick, J.: Some local properties of Gaussian vector fields. Ann. Probab. 6, 984–994 (1978)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Cuzick, J.: Multiple points of a Gaussian vector field. Z. Wahrsch. Verw. Gebiete. 61, 431–436 (1982)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Dalang, R.C.: Extending martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4(6), 1–29 (1999) (Erratum in Electron. J. Probab. 6(6), 1–5 (2001))Google Scholar
  21. 21.
    Dalang, R.C., Khoshnevisan, D., Nualart, E.: Hitting probabilities for systems of non-linear stochastic heat equations with additive noise. Latin Am. J. Probab. Statist. (Alea) 3, 231–271 (2007)Google Scholar
  22. 22.
    Dalang, R.C., Khoshnevisan, D., Nualart, E.: Hitting probabilities for the non-linear stochastic heat equation with multiplicative noise. Probab. Theor. Relat. Fields 117, 371–427 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Dalang, R.C., Khoshnevisan, D., Nualart, E., Wu, D., Xiao, Y.: Critical Brownian sheet does not have double points. Ann. Probab. 40, 1829–1859 (2012)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Dalang, R.C., Sanz-Solé, M.: Criteria for hitting probabilities with applications to systems of stochastic wave equations. Bernoulli 16, 1343–1368 (2010)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Dembo, A., Peres, P., Rosen, J., Zeitouni, O.: Thick points for spatial Brownian motion: multifractal analysis of occupation measure. Ann. Probab. 28, 1–35 (2000)MathSciNetMATHGoogle Scholar
  26. 26.
    Didier, G., Pipiras, V.: Integral representations of operator fractional Brownian motions. Bernoulli 17, 1–33 (2011)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Dindar, Z.: On the Hausdorff dimension of the set generated by exceptional oscillations of a two-parameter Wiener process. J. Multivar. Anal. 79, 52–70 (2001)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Ehm, E.: Sample function properties of multi-parameter stable processes. Z. Wahrsch. Verw. Gebiete 56, 195–228 (1981)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Estrade, A., Wu, D., Xiao, Y.: Packing dimension results for anisotropic Gaussian random fields. Commun. Stoch. Anal. 5, 41–64 (2011)MathSciNetGoogle Scholar
  30. 30.
    Falconer, K.J.: Fractal Geometry—Mathematical Foundations and Applications, 2nd edn. Wiley, NJ (2003)MATHCrossRefGoogle Scholar
  31. 31.
    Falconer, K.J., Howroyd, J.D.: Packing dimensions of projections and dimension profiles. Math. Proc. Cambridge Philos. Soc. 121, 269–286 (1997)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Falconer, K.J., Xiao, Y.: Generalized dimensions of images of measures under Gaussian processes (2011). SubmittedGoogle Scholar
  33. 33.
    Fouché, W.L., Mukeru, S.: Fourier structure of the zero set of Brownian motion (2011). PreprintGoogle Scholar
  34. 34.
    Geman, D., Horowitz, J.: Occupation densities. Ann. Probab. 8, 1–67 (1980)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Gneiting, T.: Nonseparable, stationary covariance functions for space-time data. J. Am. Statist. Assoc. 97, 590–600 (2002)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Goldman, A.: Points multiples des trajectoires de processus gaussiens. Z. Wahrsch. Verw. Gebiete 57, 481–494 (1981)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Hawkes, J.: Measures of Hausdorff type and stable processes. Mathematika 25, 202–212 (1978)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Howroyd, J.D.: Box and packing dimensions of projections and dimension profiles. Math. Proc. Cambridge Phil. Soc. 130, 135–160 (2001)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Hwang, K.-S., Wang, W., Xiao, Y.: The modulus of non-differentiability of a fractional Brownian motion (2012). PreprintGoogle Scholar
  40. 40.
    Hu, H., Nualart, D.: Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33, 948–983 (2005)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Hu, Y., Nualart, D.: Stochastic heat equation driven by fractional noise and local time. Probab. Theor. Relat. Fields 143, 285–328 (2009)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Istas, J.: Spherical and hyperbolic fractional Brownian motion. Electron. Comm. Probab. 10, 254–262 (2005)MathSciNetMATHGoogle Scholar
  43. 43.
    Kahane, J.-P.: Some Random Series of Functions, 2nd edn. Cambridge University Press, Cambridge (1985a)MATHGoogle Scholar
  44. 44.
    Kahane, J.-P.: Ensembles aleatoires et dimensions. In: Recent Progress in Fourier Analysis (El Escorial, 1983), pp. 65–121. North-Holland, Amsterdam (1985b)Google Scholar
  45. 45.
    Kahane, J.-P., Salem, R.: Ensembles Parfaits et Series Trigonometriques, 2nd edn. Hermann, Paris (1994)MATHGoogle Scholar
  46. 46.
    Kamont, A.: On the fractional anisotropic Wiener field. Probab. Math. Statist. 16, 85–98 (1996)MathSciNetMATHGoogle Scholar
  47. 47.
    Kaufman, R.: Une propriété métrique du mouvement brownien. C. R. Acad. Sci. Paris 268, 727–728 (1968)MathSciNetGoogle Scholar
  48. 48.
    Kaufman, R.: Measures of Hausdorff-type, and Brownian motion. Mathematika 19, 115–119 (1972)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Kaufman, R.: Large increments of Brownian motion. Nagoya Math. J. 56, 139–145 (1975)MathSciNetMATHGoogle Scholar
  50. 50.
    Kaufman, R.: Temps locaux et dimensions. C. R. Acad. Sci. Paris Sér. I Math. 300, 281–282 (1985)MathSciNetMATHGoogle Scholar
  51. 51.
    Khoshnevisan, D.: Multiparameter Processes: An Introduction to Random Fields. Springer, New York (2002)MATHGoogle Scholar
  52. 52.
    Khoshnevisan, D.: Intersections of Brownian motions. Expos. Math. 21, 97–114 (2003)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Khoshnevisan, D., Peres, Y., Xiao, Y.: Limsup random fractals. Electron. J. Probab. 5(4), 1–24 (2000)MathSciNetGoogle Scholar
  54. 54.
    Khoshnevisan, D., Schilling, R., Xiao, Y.: Packing dimension profiles and Lévy processes. Bull. London Math. Soc. (2012). doi:10.1112/blms/bds022MathSciNetGoogle Scholar
  55. 55.
    Khoshnevisan, D., Shi, Z.: Brownian sheet and capacity. Ann. Probab. 27, 1135–1159 (1999)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Khoshnevisan, D., Wu, D., Xiao, Y.: Sectorial local non-determinism and the geometry of the Brownian sheet. Electron. J. Probab. 11, 817–843 (2006)MathSciNetGoogle Scholar
  57. 57.
    Khoshnevisan, D., Xiao, Y.: Images of the Brownian sheet. Trans. Am. Math. Soc. 359, 3125–3151 (2007)MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    Khoshnevisan, D., Xiao, Y.: Packing dimension of the range of a Lévy process. Proc. Am. Math. Soc. 136, 2597–2607 (2008a)MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    Khoshnevisan, D., Xiao, Y.: Packing dimension profiles and fractional Brownian motion. Math. Proc. Cambridge Philos. Soc. 145, 205–213 (2008b)MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Khoshnevisan, D., Xiao, Y.: Brownian motion and thermal capacity (2012). SubmittedGoogle Scholar
  61. 61.
    Kõno, N.: Double points of a Gaussian sample path. Z. Wahrsch. Verw. Gebiete 45, 175–180 (1978)MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Le Gall, J.-F.: The exact Hausdorff measure of Brownian multiple points. In: Cinlar, E., Chung, K.L., Getoor, R.K. (eds.) Seminar on Stochastic Processes (Charlottesville, Va., 1986), pp. 107–137. Progress in Probability and Statistics, vol. 13. Birkhäuser, Boston (1987a)Google Scholar
  63. 63.
    Le Gall, J.-F.: Temps locaux d’intersection et points multiples des processus de Lévy. In: Séminaire de Probabilités, XXI. Lecture Notes in Mathematics, vol. 1247, pp. 341–374. Springer, Berlin (1987b)Google Scholar
  64. 64.
    Le Gall, J.-F.: The exact Hausdorff measure of Brownian multiple points II. In: Seminar on Stochastic Processes (Gainesville, FL, 1988), pp. 193–197. Progress in Probability, vol. 17. Birkhäuser, Boston (1989)Google Scholar
  65. 65.
    Le Gall, J.-F., Taylor, S.J.: The packing measure of planar Brownian motion. In: Seminar on Stochastic Processes (Charlottesville, Va., 1986), pp. 139–147. Progress in Probability and Statistics, vol. 13. Birkhäuser, Boston (1987)Google Scholar
  66. 66.
    Lévy, P.: La mesure de Hausdorff de la courbe du mouvement brownien. Giorn. Ist. Ital. Attuari. 16, 1–37 (1953)MathSciNetMATHGoogle Scholar
  67. 67.
    Li, W.V., Shao, Q.-M.: Gaussian processes: inequalities, small ball probabilities and applications. In: Rao, C.R., Shanbhag, D. (eds.) Stochastic Processes: Theory and Methods. Handbook of Statistics, vol. 19, pp. 533–597. North-Holland, Amsterdam (2001)CrossRefGoogle Scholar
  68. 68.
    Li, Y., Xiao, Y.: Multivariate operator-self-similar random fields. Stoch. Process. Appl. 121, 1178–1200 (2011)MathSciNetMATHCrossRefGoogle Scholar
  69. 69.
    Lin, H.: The local times and Hausdorff measure for level sets of a Wiener sheet. Sci. China Ser. A 44, 696–708 (2001)MathSciNetMATHGoogle Scholar
  70. 70.
    Luan, N., Xiao, Y.: Chung’s law of the iterated logarithm for anisotropic Gaussian random fields. Statist. Probab. Lett. 80, 1886–1895 (2010)MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    Luan, N., Xiao, Y.: Spectral conditions for strong local nondeterminism and exact Hausdorff measure of ranges of Gaussian random fields. J. Fourier Anal. Appl. 18, 118–145 (2012)MathSciNetMATHCrossRefGoogle Scholar
  72. 72.
    Marcus, M.B., Rosen, J.: Markov Processes, Gaussian Processes, and Local Times. Cambridge University Press, Cambridge (2006)MATHCrossRefGoogle Scholar
  73. 73.
    Mason, D.M., Shi, Z.: Small deviations for some multi-parameter Gaussian processes. J. Theor. Probab. 14, 213–239 (2001)MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    Mason, D.J., Xiao, Y.: Sample path properties of operator self-similar Gaussian random fields. Theor. Probab. Appl. 46, 58–78 (2002)MathSciNetCrossRefGoogle Scholar
  75. 75.
    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)MATHCrossRefGoogle Scholar
  76. 76.
    Meerschaert, M.M., Wang, W., Xiao, Y.: Fernique-type inequalities and moduli of continuity of anisotropic Gaussian random fields. Trans. Am. Math. Soc. 365, 1081–1107 (2013)MathSciNetCrossRefGoogle Scholar
  77. 77.
    Mockenhaupt, G.: Salem sets and restriction properties of Fourier transforms. Geom. Funct. Anal. 10, 1579–1587 (2000)MathSciNetMATHCrossRefGoogle Scholar
  78. 78.
    Monrad, D., Pitt, L.D.: Local nondeterminism and Hausdorff dimension. In: Cinlar, E., Chung, K.L., Getoor, R.K. (eds.) Progress in Probability and statistics. Seminar on Stochastic Processes 1986, pp. 163–189. Birkhäuser, Boston (1987)CrossRefGoogle Scholar
  79. 79.
    Monrad, D., Rootzén, H.: Small values of Gaussian processes and functional laws of the iterated logarithm. Probab. Theor. Relat. Fields 101, 173–192 (1995)MATHCrossRefGoogle Scholar
  80. 80.
    Mörters, P., Shieh, N.-R.: The exact packing measure of Brownian double points. Probab. Theor. Relat. Fields 143, 113–136 (2009)MATHCrossRefGoogle Scholar
  81. 81.
    Mountford, T.S.: Uniform dimension results for the Brownian sheet. Ann. Probab. 17, 1454–1462 (1989)MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    Mueller, C., Tribe, R.: Hitting probabilities of a random string. Electron. J. Probab. 7(10), 1–29 (2002)MathSciNetGoogle Scholar
  83. 83.
    Nualart, E., Viens, F.: The fractional stochastic heat equation on the circle: time regularity and potential theory. Stoch. Process. Appl. 119, 1505–1540 (2009)MathSciNetMATHCrossRefGoogle Scholar
  84. 84.
    Oksendal, B., Zhang, T.: Multiparameter fractional Brownian motion and quasi-linear stochastic partial differential equations. Stoch. Stoch. Rep. 71, 141–163 (2000)MathSciNetGoogle Scholar
  85. 85.
    Orey, S., Pruitt, W.E.: Sample functions of the N-parameter Wiener process. Ann. Probab. 1, 138–163 (1973)MathSciNetMATHCrossRefGoogle Scholar
  86. 86.
    Orey, S., Taylor, S.J.: How often on a Brownian path does the law of the iterated logarithm fail? Proc. London Math. Soc. 28, 174–192 (1974)MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    Perkins, E.A.: The exact Hausdorff measure of the level sets of Brownian motion. Z. Wahrsch. Verw. Gebiete 58, 373–388 (1981)MathSciNetMATHCrossRefGoogle Scholar
  88. 88.
    Perkins, E.A., Taylor, S.J.: Uniform measure results for the image of subsets under Brownian motion. Probab. Theor. Relat. Fields 76, 257–289 (1987)MathSciNetMATHCrossRefGoogle Scholar
  89. 89.
    Pitt, L.D.: Local times for Gaussian vector fields. Indiana Univ. Math. J. 27, 309–330 (1978)MathSciNetMATHCrossRefGoogle Scholar
  90. 90.
    Rosen, J.: Self-intersections of random fields. Ann. Probab. 12, 108–119 (1984)MathSciNetMATHCrossRefGoogle Scholar
  91. 91.
    Rosen, J.: The intersection local time of fractional Brownian motion in the plane. J. Multivar. Anal. 23, 37–46 (1987)MATHCrossRefGoogle Scholar
  92. 92.
    Samorodnitsky, G., Taqqu, M.S.: Stable non-Gaussian processes: stochastic models with infinite variance. Chapman and Hall, (1994)Google Scholar
  93. 93.
    Shieh, N.-R., Xiao, Y.: Images of Gaussian random fields: Salem sets and interior points. Studia Math. 176, 37–60 (2006)Google Scholar
  94. 94.
    Shieh, N.-R., Xiao, Y.: Hausdorff and packing dimensions of the images of random fields. Bernoulli 16, 926–952 (2010)MathSciNetMATHCrossRefGoogle Scholar
  95. 95.
    Stein, M.L.: Space-time covariance functions. J. Am. Statist. Assoc. 100, 310–321 (2005)MATHCrossRefGoogle Scholar
  96. 96.
    Talagrand, M. The small ball problem for the Brownian sheet. Ann. Probab. 22, 1331–1354 (1994)MathSciNetMATHCrossRefGoogle Scholar
  97. 97.
    Talagrand, M.: Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann. Probab. 23, 767–775 (1995)MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    Talagrand, M.: Multiple points of trajectories of multiparameter fractional Brownian motion. Probab. Theor. Relat. Fields 112, 545–563 (1998)MathSciNetMATHCrossRefGoogle Scholar
  99. 99.
    Talagrand, M.: Generic Chaining. Springer, New York (2006)Google Scholar
  100. 100.
    Talagrand, M., Xiao, Y.: Fractional Brownian motion and packing dimension. J. Theor. Probab. 9, 579–593 (1996)MathSciNetMATHCrossRefGoogle Scholar
  101. 101.
    Taylor, S.J.: The Hausdorff α-dimensional measure of Brownian paths in n-space. Proc. Cambridge Philo. Soc. 49, 31–39 (1953)MATHCrossRefGoogle Scholar
  102. 102.
    Taylor, S.J.: The measure theory of random fractals. Math. Proc. Cambridge Philo. Soc. 100, 383–406 (1986)MATHCrossRefGoogle Scholar
  103. 103.
    Taylor, S.J., Tricot, C.: Packing measure and its evaluation for a Brownian path. Trans. Am. Math. Soc. 288, 679–699 (1985)MathSciNetMATHCrossRefGoogle Scholar
  104. 104.
    Testard, F.: Polarité, points multiples et géométrie de certains processus gaussiens. Publ. du Laboratoire de Statistique et Probabilités de l’U.P.S. Toulouse, 01–86 (1986)Google Scholar
  105. 105.
    Tindel, S., Tudor, C.A., Viens, F.: Sharp Gaussian regularity on the circle, and applications to the fractional stochastic heat equation. J. Funct. Anal. 217, 280–313 (2004)MathSciNetMATHCrossRefGoogle Scholar
  106. 106.
    Tricot, C.: Two definitions of fractional dimension. Math. Proc. Cambridge Philo. Soc. 91, 57–74 (1982)MathSciNetMATHCrossRefGoogle Scholar
  107. 107.
    Walsh, J.B.: Propagation of singularities in the Brownian sheet. Ann. Probab. 10, 279–288 (1982)MathSciNetMATHCrossRefGoogle Scholar
  108. 108.
    Wang, W.: Almost-sure path properties of fractional Brownian sheet. Ann. Inst. H. Poincar\(\acute{e}\) Probab. Statist. 43, 619–631 (2007)Google Scholar
  109. 109.
    Wu, D., Xiao, Y.: Geometric properties of the images fractional Brownian sheets. J. Fourier Anal. Appl. 13, 1–37 (2007)MathSciNetMATHCrossRefGoogle Scholar
  110. 110.
    Wu, D., Xiao, Y.: Uniform Hausdorff dimension results for Gaussian random fields. Sci. in China, Ser. A 52, 1478–1496 (2009)MathSciNetMATHGoogle Scholar
  111. 111.
    Wu, D., Xiao, Y.: Regularity of intersection local times of fractional Brownian motions. J. Theor. Probab. 23, 972–1001 (2010)MathSciNetMATHCrossRefGoogle Scholar
  112. 112.
    Wu, D., Xiao, Y.: On local times of anisotropic Gaussian random fields. Comm. Stoch. Anal. 5, 15–39 (2011)MathSciNetGoogle Scholar
  113. 113.
    Xiao, Y.: Dimension results for Gaussian vector fields and index-α stable fields. Ann. Probab. 23, 273–291 (1995)MathSciNetMATHCrossRefGoogle Scholar
  114. 114.
    Xiao, Y.: Packing measure of the sample paths of fractional Brownian motion. Trans. Amer. Math. Soc. 348, 3193–3213 (1996)MathSciNetMATHCrossRefGoogle Scholar
  115. 115.
    Xiao, Y.: Hausdorff dimension of the graph of fractional Brownian motion. Math. Proc. Cambridge Philo. Soc. 122, 565–576 (1997)MATHCrossRefGoogle Scholar
  116. 116.
    Xiao, Y.: Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theor. Relat. Fields 109, 129–157 (1997)MATHCrossRefGoogle Scholar
  117. 117.
    Xiao, Y.: Packing dimension of the image of fractional Brownian motion. Statist. Probab. Lett. 333, 379–387 (1997)CrossRefGoogle Scholar
  118. 118.
    Xiao, Y.: Hausdorff-type measures of the sample paths of fractional Brownian motion. Stoch. Process. Appl. 74, 251–272 (1998)MATHCrossRefGoogle Scholar
  119. 119.
    Xiao, Y.: Hitting probabilities and polar sets for fractional Brownian motion. Stoch. Stoch. Rep. 66, 121–151 (1999)MATHGoogle Scholar
  120. 120.
    Xiao, Y.: The packing measure of the trajectories of multiparameter fractional Brownian motion. Math. Proc. Cambridge Philo. Soc. 135, 349–375 (2003)MATHCrossRefGoogle Scholar
  121. 121.
    Xiao, Y.: Random fractals and Markov processes. In: Lapidus, M.L., van Frankenhuijsen, M. (eds.) Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, pp. 261–338. American Mathematical Society, New York (2004)CrossRefGoogle Scholar
  122. 122.
    Xiao, Y.: Strong local nondeterminism and the sample path properties of Gaussian random fields. In: Lai, T.L, Shao, Q., Qian, L. (eds.) Asymptotic Theory in Probability and Statistics with Applications, pp. 136–176. Higher Education Press, Beijing (2007)Google Scholar
  123. 123.
    Xiao, Y.: Sample path properties of anisotropic Gaussian random fields. In: Khoshnevisan, D., Rassoul-Agha, F. (eds.) A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1962, pp. 145–212. Springer, New York (2009)CrossRefGoogle Scholar
  124. 124.
    Xiao, Y.: A packing dimension theorem for Gaussian random fields. Statist. Probab. Lett. 79, 88–97 (2009)MathSciNetMATHCrossRefGoogle Scholar
  125. 125.
    Xiao, Y.: Properties of strong local nondeterminism and local times of stable random fields. In: Dalang, R.C., Dozzi, M., Russo, F. (eds.) Seminar on Stochastic Analysis, Random Fields and Applications VI, Progress in Probability, vol. 63, pp. 279–310. Birkhäuser, Basel (2011)CrossRefGoogle Scholar
  126. 126.
    Xiao, Y., Zhang, T.: Local times of fractional Brownian sheet. Probab. Theor. Relat. Fields 124, 204–226 (2002)MathSciNetMATHCrossRefGoogle Scholar
  127. 127.
    Xue, Y., Xiao, Y.: Fractal and smoothness properties of anisotropic Gaussian models. Frontiers Math. China 6, 1217–1246 (2011)MathSciNetCrossRefGoogle Scholar
  128. 128.
    Zhang, L.X.: Two different kinds of liminfs on the LIL for two-parameter Wiener processes. Stoch. Process. Appl. 63, 175–188 (1996)CrossRefGoogle Scholar
  129. 129.
    Zimmerman, G.J.: Some sample function properties of the two-parameter Gaussian process. Ann. Math. Statist. 43, 1235–1246 (1972)MathSciNetMATHCrossRefGoogle Scholar
  130. 130.
    Zygmund, A.: Trigonometric Series, vol. I. Cambridge University Press, Cambridge (1959)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA

Personalised recommendations