Acta mathematica scientia, Series B >
POLAR SETS OF MULTIPARAMETER BIFRACTIONAL BROWNIAN SHEETS
Received date: 2007-10-08
Revised date: 2008-01-27
Online published: 2010-05-20
Supported by
Research supported by the national natural foundation of China (70871104), the key research base for humanities and social sciences of Zhejiang Provincial high education talents (Statistics of Zhejiang Gongshang University)
Let BH, K=BH, K(t), t ∈ RN+ } be an (N, d)-bifractional Brownian sheet with Hurst indices H=(H1,…, HN) ∈ (0, 1)N and K=(K1, …, KN) ∈ (0, 1]N. The properties of the polar sets of BH, K are discussed. The sufficient conditions and necessary conditions for a compact set to be polar for BH, K are proved. The infimum of Hausdorff dimensions of its non-polar sets are obtained by means of constructing a Cantor-type set to connect its Hausdorff dimension and capacity.
CHEN Zhen-Long , LI Hui-Qiong . POLAR SETS OF MULTIPARAMETER BIFRACTIONAL BROWNIAN SHEETS[J]. Acta mathematica scientia, Series B, 2010 , 30(3) : 857 -872 . DOI: 10.1016/S0252-9602(10)60084-7
[1] Addie R, Mannersalo P, Norros I. Performance formulae for queues with Gaussian input. European Trans Telecommunications, 2002, 13(3): 183--196
[2] Adler R J. The Geomentry of Random Fields. New York: Wiley, 1981
[3] Anh V V, Angulo J M, Ruiz-Medina M D. Possible long-range dependence in fractional random fields.J Statist Plann. Inference, 1999, 80: 95--110
[4] Benassi A, Bertrand P, Istas J. Identification of the hurst exponent of a step multifractional Brownian motion. Statistical Inference for Stochastic Processes, 2000, 13: 101--111
[5] Benson D A, Meerschaert M M, Baeumer B. Aquifer operator-scaling and the effect on solute mixing and dispersion. Water Resour Res, 2006, 42: W01415
[6] Bonami A, Estrade A. Anisotropic analysis of some Gaussian models. J Fourier Anal Appl, 2003, 9: 215--236
[7] Chen Zhenlong. The properties of the polar sets for Brownian sheet (in Chinese). J Math (PRC), 1997, 15: 373--378
[8] Cheridito P. Gaussian moving averages, semimartingales and option pricing. Stochastic Process Appl, 2004, 109: 47--68
[9] Hawkes J. Measures of Hausdorff type and stable processes. Mathematika, 1978, 25: 202--212
[10] Houdre C, Villa J. An example of infinite dimensional quasi-helix. Stochastic models (Mexico City), 2002: 195-201, Contemp Math, 2003, 336, Amer Math Soc, Providence, RI
[11] Kahane J P. Points multiples des processus de lévy symétriques restreints â un ensemble de valurs du temps. Orsay: Sém Anal Harm, 1983, 38(2): 74--105
[12] Kahane J P. Some Random Series of Functions. 2nd ed. Cambridge University Press, 1985
[13] Kakutani S. Two-dimensional Brownian motion and harmonic functions. Proc Imperial Acad, 1944, 20: 706--714
[14] Khoshnevisan D. Some polar sets for the Brownian sheet. Sém de Prob XXXI. Lecture Notes in Mathematics, 1997, 1655: 190--197
[15] Mannersalo P, Norros I. A most probable path approach to queueing systems with general Gaussian input. Comp Networks, 2002, 40(3): 399--412
[16] Port S C, Stone C J. Brownian Motion and Classical Potential Theory. New York: Academic Press, 1978
[17] Rogers C A. Hausdorff Measures. London: Cambridge University Press, 1970
[18] Russo F, Tudor C A. On the bi-fractional Brownian motion. Stoch Process Appl, 2006, 5: 830--856
[19] Taylor S J, Tricot C. Packing measure and its evaluation for a Brownian path. Trans Amer Math Soc, 1985, 288: 679--699
[20] Taylor S J, Watson N A. A Hausdorff measure classification of polar sets for the heat equation. Math Proc Camb Philos Soc, 1985, 97: 325--344
[21] Testard F. Quelques propri\'{e}t\'es g\'eom\'etriques de certains processus gaussiens. C R Acad Sc Paris, 1985, 300, Série I : 497--500
[22] Testard F. Dimension asym\'etrique et ensembles doublement non polairs. C R Acad Sc Paris, 1986a, 303, Série I: 579--581
[23] Testard F. Polarit\'e, points multiples et géométrie de certain processus gaussiens. Toulouse: Publ du Laboratoire de Statistique et Probabilit\'es de l' U P S, 1986b, mars: 1--86
[24] Tudor C A, Xiao Y. Sample path properties of bifractional Brownian motion. Bernoulli, 2007, 13: 1023--1052
[25] Xiao Y. Hitting probabilities and polar sets for fractional Brownian motion. Stochastics and Stochastics Reports, 1999, 66: 121--151
[26] Xiao Y. Sample path properties of anisotropic Gaussian random fields//Khoshnevisan D, Rassoul-Agha F. A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Math, 1962. New York: Springer, 2009: 145--212
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