Smoothness of local times and self-intersection local times of Gaussian random fields

Abstract

This paper is concerned with the smoothness (in the sense of Meyer- Watanabe) of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.

Keywords

Anisotropic Gaussian field / local time / collision local time / intersection local time / self-intersection local time / chaos expansion

References

[1] Ayache A, Wu D, Xiao Y. Joint continuity of the local times of fractional Brownian sheets. Ann Inst Henri Poincaré Probab Stat, 2008, 44: 727-748

[2] Ayache A, Xiao Y. Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets. J Fourier Anal Appl, 2005, 11: 407-439

[3] Biermé H, Lacaux C, Xiao Y. Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields. Bull Lond Math Soc, 2009, 41: 253-273

[4] Chen C, Yan L. Remarks on the intersection local time of fractional Brownian motions. Statist Probab Lett, 2011, 81: 1003-1012

[5] Chen Z, Xiao Y. On intersections of independent anisotropic Gaussian random fields. Sci China Math, 2012, 55: 2217-2232

[6] Dalang R C, Mueller C, Xiao Y. Polarity of points for a wide class of Gaussian random fields. Preprint, 2015

[7] Eddahbi M, Lacayo R, Solé J L, Vives J, Tudor C A. Regularity of the local time for the d-dimensional fractional Brownian motion with N-parameters. Stoch Anal Appl, 2005, 23: 383-400

[8] Eddahbi M, Lacayo R, Solé J L, Vives J, Tudor C A. Renormalization of the local time for the d-dimensional fractional Brownian motion with N parameters. Nagoya Math J, 2007, 186: 173-191

[9] Eddahbi M, Vives J. Chaotic expansion and smoothness of some functionals of the fractional Brownian motion. J Math Kyoto Univ, 2003, 43: 349-368

[10] Geman D, Horowitz J. Occupation densities. Ann Probab, 1980, 8: 1-67

[11] Hu Y. Self-intersection local time of fractional Brownian motion—via chaos expansion. J Math Kyoto Univ, 2001, 41: 233-250

[12] Hu Y, Øksendal B. Chaos expansion of local time of fractional Brownian motions. Stoch Anal Appl, 2002, 20: 815-837

[13] Hu Y, Nualart D. Renormalized self-intersection local time for fractional Brownian motion. Ann Probab, 2005, 33: 948-983

[14] Imkeller P, Perez-Abreu V, Vives J. Chaos expansion of double intersection local time of Brownian motion in ℝd and renormalization. Stochastic Process Appl, 1995, 56: 1-34

[15] Imkeller P, Weisz F. The asymptotic behaviour of local times and occupation integrals of the N-parameter Wiener process in ℝd. Probab Theory Related Fields, 1994, 98: 47-75

[16] Imkeller P, Weisz F. Critical dimensions for the existence of self-intersection local times of the Brownian sheet in ℝd. In: Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993). Progr Probab, 36. Basel: Birkhäuser, 1995, 151-168

[17] Imkeller P, Weisz F. Critical dimensions for the existence of self-intersection local times of the N parameter Brownian motion in ℝd. J Theoret Probab, 1999, 12: 721-737

[18] Jiang Y, Wang Y. Self-intersection local times and collision local times of bifractional Brownian motions. Sci China Math, 2009, 52: 1905-1919

[19] Meyer P A. Quantum for Probabilists. Lecture Notes in Math, Vol 1538. Heidelberg: Springer, 1993

[20] Mueller C, Tribe R. Hitting properties of a random string. Electron J Probab, 2002, 7: 1-29

[21] Nualart D. The Malliavin Calculus and Related Topics. New York: Springer, 2006

[22] Nualart D, Vives J. Chaos expansion and local time. Publ Mat, 1992, 36: 827-836

[23] Pitt L P. Local times for Gaussian vector fields. Indiana UnivMath J, 1978, 27: 309-330

[24] Shen G, Yan L. Smoothness for the collision local times of bifractional Brownian motions. Sci China Math, 2011, 54: 1859-1873

[25] Shen G, Yan L, Chen C. Smoothness for the collision local time of two multidimensional bifractional Brownian motions. Czechoslovak Math J, 2012, 62: 969-989

[26] Tudor C, Xiao Y. Sample paths of the solution to the fractional-colored stochastic heat equation. Preprint, 2015

[27] Watanabe S. Stochastic Differential Equation and Malliavian Calculus. Tata Institute of Fundamental Research. Berlin: Springer, 1984

[28] Wu D, Xiao Y. Fractal properties of the random string process. IMS Lecture Notes Monogr Ser–High Dimensional Probability, 2006, 51: 128-147

[29] Wu D, Xiao Y. Geometric properties of the images of fractional Brownian sheets. J Fourier Anal Appl, 2007, 13: 1-37

[30] Wu D, Xiao Y. Regularity of intersection local times of fractional Brownian motions. J Theoret Probab, 2010, 23: 972-1001

[31] Wu D, Xiao Y. On local times of anisotropic Gaussian random fields. Commun Stoch Anal, 2011, 5: 15-39

[32] 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 Math, Vol 1962. New York: Springer, 2009, 145-212

[33] Xiao Y, Zhang T. Local times of fractional Brownian sheets. Probab Theory Related Fields, 2002, 124: 204-226

[34] Yan L, Liu J, Chen C. On the collision local time of bifractional Brownian motions. Stoch Dyn, 2009, 9: 479-491

[35] Yan L, Shen G. On the collision local time of sub-fractional Brownian motions. Statist Probab Lett, 2010, 80: 296-308

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Higher Education Press and Springer-Verlag Berlin Heidelberg