独立增量随机场的分形性质
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摘要
随机分形是融概率论、经典分析和几何学于一体的新兴数学分支。随机过程样本轨道的分形性质是随机分形理论的重要组成部分和活跃的研究方向。鉴于某些独立增量随机过程的很多良好性质和结构，很有必要将其推广到更一般情形而讨论其相关的分形性质。本文主要研究几类随机过程的有关分形性质。具体内容如下：
·研究了N指标d维广义Brownian Sheet的极集的性质，得到了其极集的必要条件与充分条件。同时通过一个特殊的Cantor型集的构造将维数与容度巧妙地结合起来，得到了广义Brownian Sheet极集的Hausdorff维数的下确界。其结论解决了Xiao提出的关于布朗单的极集的维数问题[145]。此结果形式漂亮，对于布朗单也是新的。
·讨论了N指标d维广义Brownian Sheet极函数的特征，得到了满足Lipschitz条件的连续函数类与广义Brownian Sheet的极函数类之间的关系，给出了广义BrownianSheet不动点的Hausdorff维数和Kolmogorov下熵指数的一个不等式。同时解决了LeGall提出的关于布朗运动非极、连续的H(?)lder函数存在性问题[54]。
·讨论了N指标d维广义Brownian Sheet象集的有限项代数和的Hausdorff维数、Packing维数、Lebesgue测度及内点的存在性。解决了Mountford提出的关于布朗单象集的内点存在性问题[122]．
·研究了N指标d维广义Brownian Sheet的容度问题。利用鞅的方法讨论了广义布朗单的碰撞概率与容度之间的关系，给出了其碰撞概率的容度估计。同时也讨论了广义布朗单的象集的Lebesgue测度与Bessel-Riesz容度之间的关系，给出了其象集的Bessel-Riesz容度估计。这些结果包含了Brownian Sheet和可加布朗运动的结果，解决了J P Kahane提出的关于可加布朗运动的Bessel-Riesz容度问题[62]。
·讨论了N指标d维广义α-stable过程的自相交局部时的存在性和联合连续性问题。给出了其自相交局部时平方可积和联合连续的一个充分条件，并通过对广义α-stable过程自相交局部时的H(?)lder条件的讨论，证明了其重点的存在性，得到了广义α-stable过程多重时的Hausdorff维数及测度的下界。
·利用N指标d维广义α-stable过程在区域上增量最大值的尾概率分布和逗留时分布，得到了未必具有随机一致H(?)lder条件的广义α-stable过程象集、图集的一致Hausdorff维数和一致Packing维数的上、下界。
·研究了某些具有强局部不确定性的Gaussian随机场的Hausdorff测度和Hausdorff型测度。给出了其图集的确切Hausdorff测度，象集和图集的确切Hausdorff型测度及其估计，证明了象集和图集的Hausdorff测度函数仍是Hausdorff型测度函数。在一定条件下，通过一些更小的测度函数给出了象集和图集的有限正的Hausdorff型测度。
西安电子科技大学博士学位论文:独立增t随机场的分形性质
·研究了分式布朗运动的Packing型测度问题.给出了Packing型测度的定义及性质，
得到了分式Brownian运动的象集和图集的Packing型测度的上、下界.同时也建立
了分式Browllian运动逗留时的hminf型重对数律.
·研究了非退化扩散过程图集、水平集和极集的Hausd。甫维数.给出了比文献【147}
更一般的极性的充分条件，并通过Cantor型集的构造法，得到了既无高斯性，也
无平稳和独立增量性的非退化扩散过程极集的Hausdo甫维数的下确界.同时采用
Testard的方法给出了紧集上非退化扩散过程水平集的Hausdorff维数.
关键词
Hausdorff维数
广义Brownian Sheet
广义a一stable
过程极集容度
paeking型测度
极函数
paeking维数Hausdor仔型测度
Random fractal, which involves probability, classical analysis and geometry, is a new mathematics branch. The fractal properties of the sample path of stochastic processes, which are an important component of the random fractal theory, have become one of the most important and active research fields of random fractal. With many favorable properties and structures emerging in some independent increment stochastic processes, it is necessary to generalize them and discuss their related fractal properties. The purpose of this thesis is to investigate the fractal properties of several classes of stochastic processes and the main contents are the following:· The properties are studied of the polar sets for N-parameter d-dimensional generalized Brownian Sheet. The necessary conditions and sufficient conditions for a set to be polar are listed. Finally, the infimum is acquired of Hausdorff dimensions of its polar sets by means of constructing a Cantor-type set to connect Hausdorff dimension and capacity.A problem proposed by Xiao about the dimensions of its polar sets is solved[145].· The characteristics about the polar-functions for N-parameter d-dimensional generalized Brownian Sheet, are discussed. The relationship between the class of continuous functions satisfying Lipschitz condition and the class of polar-functions of generalized Brownian Sheet is obtained. The Hausdorff dimension about the stable points and the inequality about the Kolmogorov's entropy index for N-parameter d-dimensional generalized Brownian Sheet are presented and a question proposed by LeGall about the existence of no-polar, continuous functions statisfying the Holder condition is solved[54].· The following are also discussed about the Hausdorff dimension, the Packing dimension and the existence of the positive Lebesgue measure and the interior points of m-term algebraic sum of the image for generalized Brownian Sheet. A problem proposed by Mountford about the existence of the interior points of the image for Brownian Sheet is also solved[122].· The relationship between the hitting probabilities for generalized Brownian Sheet and the capacity is studied by means of multiparameter martingales. A capacity estimate for its hitting probabilities is provided. Finally the connections between the Lebesgue measure for its image and Bessel-Ricsz capacity are discussed and an explicit Bessel-Riesz capacity estimate is obtained. The conclusions also include the solution to a problem proposed by J. P. Kahane about the Bessel-Riesz capacity of additive Brown motion [62].· The existence and joint continuity are studied for the self-intersection local time of N-
parameter d-dimension generalized α-stable process. A sufficient condition is presented for the existence of square integrable and joint continuity of its self-intersection local time. The existence of its multiple points is proved by means of getting Holder condition of its self-intersection local time and the lower bounds are obtained about the Hausdorff dimension and measure of its multiple times.· The problem of the uniform dimension for image set and graph set of N-parameter d-dimensional generalized a-stable process, which may not possess the uniform stochastic Holder condition, is investigated. The uniform Hausdorff dimension and Packing dimension for its image set and graph set are obtained by means of distribution of the sojourn time and the tail probability distribution for the increment maximum of stochastic process over interval.· The study also covers the usual Hausdorff measure and Hausdorff-type measure for the image set and graph set of certain locally nondeterministic Gaussian random fields. The exact Hausdorff measure of its graph set is given. The exact Hausdorff-type measures of its image set and graph set are provided and it is proved that the usual Hausdorff measure functions for its image set and graph set are still correct measure functions. Under certain conditions, the positive and finite Hausdorff-type measures for its image set and graph set arc given by some
·研究了N指标d维广义Brownian Sheet的极集的性质，得到了其极集的必要条件与充分条件。同时通过一个特殊的Cantor型集的构造将维数与容度巧妙地结合起来，得到了广义Brownian Sheet极集的Hausdorff维数的下确界。其结论解决了Xiao提出的关于布朗单的极集的维数问题[145]。此结果形式漂亮，对于布朗单也是新的。
·讨论了N指标d维广义Brownian Sheet极函数的特征，得到了满足Lipschitz条件的连续函数类与广义Brownian Sheet的极函数类之间的关系，给出了广义BrownianSheet不动点的Hausdorff维数和Kolmogorov下熵指数的一个不等式。同时解决了LeGall提出的关于布朗运动非极、连续的H(?)lder函数存在性问题[54]。
·讨论了N指标d维广义Brownian Sheet象集的有限项代数和的Hausdorff维数、Packing维数、Lebesgue测度及内点的存在性。解决了Mountford提出的关于布朗单象集的内点存在性问题[122]．
·研究了N指标d维广义Brownian Sheet的容度问题。利用鞅的方法讨论了广义布朗单的碰撞概率与容度之间的关系，给出了其碰撞概率的容度估计。同时也讨论了广义布朗单的象集的Lebesgue测度与Bessel-Riesz容度之间的关系，给出了其象集的Bessel-Riesz容度估计。这些结果包含了Brownian Sheet和可加布朗运动的结果，解决了J P Kahane提出的关于可加布朗运动的Bessel-Riesz容度问题[62]。
·讨论了N指标d维广义α-stable过程的自相交局部时的存在性和联合连续性问题。给出了其自相交局部时平方可积和联合连续的一个充分条件，并通过对广义α-stable过程自相交局部时的H(?)lder条件的讨论，证明了其重点的存在性，得到了广义α-stable过程多重时的Hausdorff维数及测度的下界。
·利用N指标d维广义α-stable过程在区域上增量最大值的尾概率分布和逗留时分布，得到了未必具有随机一致H(?)lder条件的广义α-stable过程象集、图集的一致Hausdorff维数和一致Packing维数的上、下界。
·研究了某些具有强局部不确定性的Gaussian随机场的Hausdorff测度和Hausdorff型测度。给出了其图集的确切Hausdorff测度，象集和图集的确切Hausdorff型测度及其估计，证明了象集和图集的Hausdorff测度函数仍是Hausdorff型测度函数。在一定条件下，通过一些更小的测度函数给出了象集和图集的有限正的Hausdorff型测度。
西安电子科技大学博士学位论文:独立增t随机场的分形性质
·研究了分式布朗运动的Packing型测度问题.给出了Packing型测度的定义及性质，
得到了分式Brownian运动的象集和图集的Packing型测度的上、下界.同时也建立
了分式Browllian运动逗留时的hminf型重对数律.
·研究了非退化扩散过程图集、水平集和极集的Hausd。甫维数.给出了比文献【147}
更一般的极性的充分条件，并通过Cantor型集的构造法，得到了既无高斯性，也
无平稳和独立增量性的非退化扩散过程极集的Hausdo甫维数的下确界.同时采用
Testard的方法给出了紧集上非退化扩散过程水平集的Hausdorff维数.
关键词
Hausdorff维数
广义Brownian Sheet
广义a一stable
过程极集容度
paeking型测度
极函数
paeking维数Hausdor仔型测度
Random fractal, which involves probability, classical analysis and geometry, is a new mathematics branch. The fractal properties of the sample path of stochastic processes, which are an important component of the random fractal theory, have become one of the most important and active research fields of random fractal. With many favorable properties and structures emerging in some independent increment stochastic processes, it is necessary to generalize them and discuss their related fractal properties. The purpose of this thesis is to investigate the fractal properties of several classes of stochastic processes and the main contents are the following:· The properties are studied of the polar sets for N-parameter d-dimensional generalized Brownian Sheet. The necessary conditions and sufficient conditions for a set to be polar are listed. Finally, the infimum is acquired of Hausdorff dimensions of its polar sets by means of constructing a Cantor-type set to connect Hausdorff dimension and capacity.A problem proposed by Xiao about the dimensions of its polar sets is solved[145].· The characteristics about the polar-functions for N-parameter d-dimensional generalized Brownian Sheet, are discussed. The relationship between the class of continuous functions satisfying Lipschitz condition and the class of polar-functions of generalized Brownian Sheet is obtained. The Hausdorff dimension about the stable points and the inequality about the Kolmogorov's entropy index for N-parameter d-dimensional generalized Brownian Sheet are presented and a question proposed by LeGall about the existence of no-polar, continuous functions statisfying the Holder condition is solved[54].· The following are also discussed about the Hausdorff dimension, the Packing dimension and the existence of the positive Lebesgue measure and the interior points of m-term algebraic sum of the image for generalized Brownian Sheet. A problem proposed by Mountford about the existence of the interior points of the image for Brownian Sheet is also solved[122].· The relationship between the hitting probabilities for generalized Brownian Sheet and the capacity is studied by means of multiparameter martingales. A capacity estimate for its hitting probabilities is provided. Finally the connections between the Lebesgue measure for its image and Bessel-Ricsz capacity are discussed and an explicit Bessel-Riesz capacity estimate is obtained. The conclusions also include the solution to a problem proposed by J. P. Kahane about the Bessel-Riesz capacity of additive Brown motion [62].· The existence and joint continuity are studied for the self-intersection local time of N-
parameter d-dimension generalized α-stable process. A sufficient condition is presented for the existence of square integrable and joint continuity of its self-intersection local time. The existence of its multiple points is proved by means of getting Holder condition of its self-intersection local time and the lower bounds are obtained about the Hausdorff dimension and measure of its multiple times.· The problem of the uniform dimension for image set and graph set of N-parameter d-dimensional generalized a-stable process, which may not possess the uniform stochastic Holder condition, is investigated. The uniform Hausdorff dimension and Packing dimension for its image set and graph set are obtained by means of distribution of the sojourn time and the tail probability distribution for the increment maximum of stochastic process over interval.· The study also covers the usual Hausdorff measure and Hausdorff-type measure for the image set and graph set of certain locally nondeterministic Gaussian random fields. The exact Hausdorff measure of its graph set is given. The exact Hausdorff-type measures of its image set and graph set are provided and it is proved that the usual Hausdorff measure functions for its image set and graph set are still correct measure functions. Under certain conditions, the positive and finite Hausdorff-type measures for its image set and graph set arc given by some
引文
[1] A. Dvorezky, P. Erdos and S. Kakutani. Double points of Brownian motion in n-space. Acta. Sci. Math., 1950, 12: 64-81.
[2] A. Dvorezky, P. Erdos and S. Kakutani. Multiple points of paths of Brownian motion in the plane. Bull. Res Council Isr. Sect., 1954, F3: 364-371.
[3] A. Dvorezky, P. Erdos, S. Kakutani and S. J. Taylor. Points of multiplicity of plane Brownian paths. Bull. Res. Council Isr. Sect., 1958, F7: 175-180.
[4] A. Dvorezky. P. Erdos, S. Kakutani and S. J. Taylor. Triple points of Brownian motion in 3-space. Prec. Comb. Phil. Soc.. 1957. 53: 856-862.
[5] A. Goldman. Points multiples des trajectories des processus gaussiens, Z. W.. 1981, 59: 481-494.
[6] A. Goldman. Mouvement Brownien à plusieurs paramètres: mesure de Hausdorff des trajectoires. Astérisque. 1988.
[7] B. B. Mandelbrot and J. W. Van, Ness. Fractional Brownian motions fracfioral noises and applications. SIAM Review. 1968. 10: 422-437.
[8] B. B. Mandelbrot. Stochastic models for the earth's relif, the shape and fractal dimension of the coastlines and the number-area rule for islands. Proc. Nat. Acad. Sci. USA. 1975. 72: 3825-3828.
[9] B. B. Mandelbrot. On the geometry of hornogeneons turbulence with streas on fractional dimension of the isv-surface of scalars. J. Fluid. Mec.. 1975. 72: 401-416.
[10] B. B. Mandelbrot. Fractals:from. chance and dimension. Freeman. San Francisco. 1977.
[11] B. B. Mandelbrot. The Fractal Geomentry of Nature. Freeman. San Francisco, 1982.
[12] B. E. Fristedt and S. J. Taylor. The packing measure of a aeneral subordinator. Prob Th Rel Fields. 1992. 92: 493-510.
[13] C. A. Rogers. Hausdorff measures. Cambridge University Press. 1970.
[14] C. Bishop and Y. Peres. Packing dimension and Cartesian Products. Trans. Amer. Math. Soc.. 1996. 345: 4433-4445.
[15] C. Tricot. Two definitions of fractional dimension. Math. Proc. Canbridge Phil. Soc. 1982. 91: 57-74.
[16] Chen Zhenlong. Liu Sanyang. Dimension of polar sets for Brownian Sheet. Acta Mathematica Scientia (Series B). 2003. 23B (4): 549-561.
[17] 陈振龙，徐赐文．N指标d维广义Wiener过程像集的一致维数．应用数学学报，2003．26(2)：345-358．
[18] Chen Zhenlong. Polar functions for N-paramenter d-dimensional generalized Brownian sheet. Acta. Mathematica Scientia. 2001. 21A(4): 472-479.
[19] 陈振龙，刘三阳．N指标d维广义Wiener过程象集的m项代数和的几个性质．数学物理学报，2002．22(2)：250-257．
[20] 陈振龙，刘禄勤．d维平稳高斯过程极集的必要条件．武汉大学学报，2002，48(1)：11-14．
[21] 陈振龙，刘禄勤．d维平稳高斯过程的极函数及其相关维数．武汉大学学报2002．48(3)：485-489．
[22] 陈振龙，刘三阳．广义Wiener过程极性的两个充分条件．数学杂志，2003，23(1)：102-107．
[23] 陈振龙，刘三阳．广义α-stable过程的自相交局部时．西安电子科技大学学报，2004，31(2)：310-316．
[24] 陈振龙，刘三阳．广义α-stable过程自相交局部时增量的Hlder律及重点．数学物理学报(待发表)
[25] Chen Zhenlong. The properties of the polar set for the Brownian Sheet. J. Math. (PRC). 17(1997): 373-378.
[26] D. German and J. Horowitz. Occuption densities. Ann. Probab, 1980, 8: 1-67.
[27] D. German, J. Horowitz and Jay Rosn. A local time analysis of intersections of Brownian path in the plane. Ann. Probab., 1984, 12: 86-107.
[28] D. Khoshnevisan. Brownian sheet images and Bessel-Riesz capacity. Trans Amer Math Soc, 1999, 351(7): 2607-2622.
[29] D. Khoshnevisan and Z. Shi. Brownian sheet and capacity. Ann Probab, 1999, 27(3): 1135-1159.
[30] D. Khoshnevisan anti Y. Xiao. Level sets of additive Lévy processes. Ann Probab, 2002, 30(1): 62-100.
[31] D. Khoshnevisan. Some polar sets for the Brownian sheet. Sém. de Prob. ⅩⅩⅪ, Lecture Notes in Mathematics 1655. 190-197. 1997.
[32] D. Khoshnevisan. Multi-Parameter Processes. (To publish)
[33] D. J. Mason and Yimin Xiao. Sample path properties of operator self-similar Gaussian random fields. Ter Veroyatnost i Primenen, 2001. 46: 94-116.
[34] D. Ray. Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion. Trans. Amer. Math. Soc.. 1964. 106: 436-444
[35] D. Mourad and D. L. Pitt. Local nondeterminism and Hausdorff dimension. Pro. Prob. stat.. Seminar on stachastic processes, Boston: Birkhauser, 1986.
[36] F. Testard. Quelques propriétés géométriques de certains processus gaussiens. C. R. Acad. Sc. Paris, 300. Série Ⅰ (1985), 497-500.
[37] F. Testard. Dimension asymétrique et ensembles doublement non polairs. C. R. Acad. Sc. Paris, 303, Série Ⅰ (1986). 579-581.
[38] F. Testard. Polarité. Points multiples et géométrie de certain processus gaussiens, Publ. du Laboratoire de Statistique et Probabilités de l' U. P. S. Toulouse. mars 1986, 1-86.
[39] F. Testard. Polarité. Points Multiples et Gémotrie de Certains Processus Gaussiens. Thése Doctorat soutnue à Orsay. 1987.
[40] F. Rezakhanlou and S. J. Taylor. The packing measure of the graph of a stacess. Astérisque 157-158, 1988: 341-361
[41] G. Samorodnitsky and M. S. taqqu. Stable Non-Gaussian Random Processes. New York: Chapman and Hall. 1994.
[42] H. Kesten. An iterated logarithm law for local time. Duke Math. J, 1965, 32: 447-456.
[43] H. Kesten. Hitting probabilities of single points for processes with stationary independent increments. Trans. Amer. Math. Soc., 1969, 93.
[44] Hu X. Some fractal sets determined by stable processes. Prob. Th. Rel. Fields, 1994, 100: 205-225.
[45] 胡迪鹤等．随机分形引论．武汉：武汉大学出版社，1995
[46] 胡迪鹤，刘禄勤，肖益民，等．随机分形．数学进展，1995，24(3)：193-214．
[47] 胡迪鹤．随机过程论-基础、理论、应用．武汉大学出版社，2000．
[48] 何声武，汪嘉岗，严加安．半鞅与随机分析．北京：科学出版社，1995．
[49] J. Cuzick and J. P. Dupreez. Joint continuity of Gaussian local time. Ann. Probab., 1982, 10, 810-817.
[50] J. Lamperti. Semi-stable stochastic processes. Trans. Amer. Math. Soc., 1962, 104: 62-78.
[51] J. F. LeGall and S. T. Taylor. The packing measure of planar Brownian motion. Progress in Probability and Staistics. Seminar on Stochastic Processec. Boston: Birkhauser, 1986. 139-147.
[5
[52] J. F. LeGall. The exact Hausdorff measure of Browniaa multiple points. Seminar on Stochastic Processes. Boston: Birkhauser, 1986.
[53] J. F. LeGall and S. J. Taylor. The Packing measure of planar Brownian motion. Progress in Probability and Statistics, Seminar on Stochastic Processes, Boston: Birkhauser, 1986.
[54] J. F. LeGall. Sur les function polaires pour les mouvement browien. Seminaire de Probabilites XXII LNM, 1321: 186-189.
[55] J. F. LeGall. The exact Hausdorff measure of Borwnian multiple points Ⅱ. Seminar on stochastic processes. Boston: Birkhauser, 1989.
[56] J. F. LeGall, J. S. Rosen and N. R. Shieh. Muliple points of Levy processes. Ann. Probab, 1989. 17: 503-515.
[57] J. Hawkes. Some dimension theorems for the sample functions of stable processes. Indiana Univ Math J, 1971, 20: 733-738
[58] J. Hawkes. W. E. Pruitt. Uniform dimension results for processes with independent increments. Z. Wahrsch. Verw. Geb., 1974, 28: 277-288.
[59] J. Hawkes. Measures of Hausdorff type and stable processes. Mathematika, 1978, 25: 202-212.
[60] J. Hawkes. Multiple points for symmetric Lévy processes. Math. Proc. Camb. Phil. Soc., 1978. 83: 83-90.
[61] J. Nolan. Local nondeterminism and local times for stable processes. Prob. Th. Rel. Field., 1989, 82: 386-410.
[62] J. P. Kahane. Some Random Series of Functions London. Cambridge University Press, 2nd edition. 1985.
[63] J. P. Kahane. Points multiples des processus de lévy symétriques restreints à un ensemble de valurs du temps. Sém. Anal. Harm. 1981-1982, Publ. Math. Orsay 1983, 83(2): 74-105.
[64] J. P. Kahane. Ensembles aléatoires et dimeninons. Cours au Seminaire de l'Escurial, 1983.
[65] J. Rosen. Self-intersections of random fields. Ann. Prob., 1984, 12: 108-119.
[66] J. Vuolle-Apiala and E. S. Graversen. Duality theory for self-similar Markov processes. Ann. Inst. H. Poincare Probab. Statist., 1988, 22: 376-392.
[67] K. J. Falconer, Fractal Geometry-Mathematical Foundations and Applications. Wiley & Sons, 1990.
[68] 乐成雄．Brown单的一致Packing测度及重点的维数．武汉大学硕士论文，1990．
[69] L. D. Pitt and L. T. Tran. Local sample Path properties of Gaussian fields. Ann Probab, 1979, 7: 477-493.
[70] L. D. Pitt. Local times for Gaussian vector fields. Indiana Univ Math J, 1978, 27: 309-330
[71] 林火南，庄兴无．N指标d维广义Wiener过程的点常返性．数学年刊，1992，13A(3)：344-352．
[72] 林火南．多指标稳定过程的一致维数结果．中国科学(A辑)，1999，29(5)：414-425
[73] L. L. Helms. Introduction to Potential Theory. Wiley, New York. 1969.
[74] Liu luqin and Yimin Xiao. Hausdorff dimension theorems for self-similar Markov processes. Prob. Math. Statis., 1998, 18: 369-393.
[75] Liu luqin. The Hausdorff dimension of the image, graph, and level of self-similar process. Stochastic and Quantum Mechanics, World Scientific press, 1992, 178-188.
[76] L. T. Tran. The Hausdorff dimension of the range of the N-parameter Wiener process. Ann. Prob. 1977, 5: 235-242.
[77] 乐成雄．Brown单的一致Packing测度及重点的维数．武汉大学硕士学位论文，1990．
[78] M. B. Marcus. Hder conditions for Gaussian processes with stationary increments. Trans Amer Math Soc, 1968, 134: 29-52.
[79] M. D. Donsker and S. R. S. Varadhan. On laws of the iterated lagarithm and local times. Comm. Pure Appel. Math., 1977, 30: 707-753.
[80] M. F. Vares. Local times for two-parameter Lévy processes. Stoc. Process. Appl., 1983, 15, 59-82.
[81] M. Ledoux and M. Talagrand. Probabilities in Banach Spaces. Springer Verlag, 1991.
[82] M. Talagrand and Yimin Xiao, Fracional Brownian motion and Packing dimension. J. Theort. Probab. 9(1996), 579-593.
[83] M. Talagrand. Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann Probab, 1995, 23: 767-775.
[84] M. Talagrarrd. Lower classes for fractional Brownian motion. J Theoret Probab, 1996. 9: 191-213.
[85] M. T. Barlow and J. Hawkes. Application de l'entropic métrique a la continuity des temps locaux des processus de Lévy. C. R. Acad. Sci. Paris s é r. I Math., 1985, 301: 237-239.
[86] M. T. Barlow. Necessary and sufficient conditions for the continuity of local time of Lévy processes. Ann. Probab., 1988, 16: 1389-1427.
[87] M. T. Lacey. Limit law for local times of the Brownian sheet. Prob. Th. Rel. Fields. 1990. 86: 63-85.
[88] N. C. Jain and S. E. Pruitt. The correct measure function for the graph of a transient stable process. Z wahrsch verw Gebiete. 1968. 9: 131-138
[89] N. Kno. Sur la minoration asymptotique et le cara ctère transitoire des trajectoires des fonctions aleatories gaussiennes à valeurs dans k~d. Z wahrsch verw Gebiete, 1975. 33: 95-112.
[90] N. Ikeda and S. Watanable. Stochastic Differential Equations and Diffusion Processes. New York: North-Holland, Publishing Company. 1981.
[91] N. N. Cēntsov. Wiener random fields depending several parameters. Dokl Akad Nauk S S S R (NS), 1956, 106: 607-609.
[92] P. Imkeller. Two-parameter Martingales and their quadratic variation. Lecture Notes in Mathematics, 1308, Springer, New York, 1988.
[93] P. Lévy. La mesure de Hausdorff de la courbe du mouvement brownien. Giorn Ital Attuari. 1953, 16: 1-37.
[94] P. Lévy. Processus Stochastiques et Mouvements Browniens. Gauthier-Villars, 1948.
[95] R. J. Adler. Hausdorff dimension and Gaussian fields. Ann. Probab., 1977, 5: 145-151.
[96] R. J. Adler. The uniform dimension of the level sets of a Brownian sheet. Ann. Probab., 1978, 6: 505-515.
[97] R. J. Adler. The Geomentry of Random Fields. New York: Wiley, 1981.
[98] R. Kaufman. Measures of Hausdorff type and Brownian motion. Mathematika. 1972, 19: 115-119.
[99] R. Kaufman. Une propiété métriqé du mouvement brownien. C. R. Acad. Sci. Paris. Sér A, 1969, 268: 727-728.
[100] R. M. Blumenthal and R. K. Getoor. local times for Markor processes. Z. W., 1964, 3: 50-74.
[101] S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory. Academic Press, New York, 1978.
[102] S. Orey and N. E. Pruitt. Sample function of the N parameter Wiener processes. Ann. Prob., 1973, 1: 138-163.
[103] S. E. Graversen and J. Vuolle-Apiala. α-self-similar Markowv processes. Probab. Th. Rel. Fidlds, 1986, 71: 149-158.
[104] S. E. Graversen." Polar-functions " for Brownian motion. Z. Wahrsch. Verw Gebiete. 1982,61:261-270.
[105] S. J. Taylor. The Hausdorff Q-dimensional measure of Brownian paths in n-space. Proc. Cambridge Philos. Soc, 1953, 49:31-39.
[106] S. J. Taylor. The dimensional measure of the graph and set of zeros of a Brownian path. Proc. Cambridge Philos. Soc.,1955, 51:265-274.
[107] S. J. Taylor. The exact Hausdorff measure of the sample path for planar Brownian motion. Proc. Cambridge Phil. Soc, 1964,60:253-258.
[108] S. J. Taylor. Multiple points for the sample paths of the symmetric processes. Z. W., 1966, 5: 247-264.
[109] S. J. Taylor. The measure theory of random fractal. Math. Proc. Cambridge Phil. Soc, 1986,100(3):383-406.
[110] S. J. Taylor and N. A. Watson. A Hausdorff measure classification of polar sets for the heat equation. Math.Proc.Camb. Philos. Soc, 1985. 97: 325-344.
[111] S. J. Taylor. The use of Packing measure in the analysis of random sets. Lecture Notes in Math. 1986, 1203:214-222.
[112] S. J. Taylor and C. Tricot. Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc, 1985, 288: 679-699.
[113] S. J. Sheu. Some estimates of the transition density of a nondegener ate diffusion Markov process.Ann.Probab..1991,19(2):538-561.
[114] S. Kakutani. Two-dimensional Brownian motion and harmonic functions. Proc. Imperial Acad. Tokyo 20(1944), 706-714.
[115] S. M. Berman. Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc, 1969, 137: 277-300.
[116] S. M. Berman. Gaussian Processes with stationary increments, local times and sample function properties. Ann. Statist.,1970, 41: 1260-1272.
[117] S. M. Berman. Gaussian sample functions: uniform dimension and Holder condition nowhere. Nagoya Math. J., 1972, 46: 63-88.
[118] S. M. Berman. Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J., 1973, 23: 69-94.
[119] S. Song. Inegalites relatives aux procesus d'ornstein-Uhlenbeck a n-parametres et capacite gausienne Cn,2. Seminare de Probabilities XXVII. Lecture Notes in Math. 1557:276-301. Spring, Berlin.
[120] Theory of the Integral. New York: Dover, 1964.
[121] T. S. Mountford. An extension of a result of Kahane using Brownian local times of intersections. Stochastics, 1988,23:449-464.
[122] T. S. Mountford. A relation between Hausdorff dimension and a condition on time sets for the image by the Brownian Sheet to possess interior points. Bull Lon. Math. Soc. 1989, 21: 179-185.
[123] T. S. Mountford. Uniform dimension results for the Brownian Sheet. Ann. Prob., 1989, 4:1454-1462.
[124] T. S. Salisbury. Energy.and intersection of Markov chains. In Random Discrete Structures 213- 225. New York: Springer,1995.
[125] W. Ehm. Sample function properties of multi-parameter stable, processes. Z Wahrsch Verw Geb, 1981, 56, 195-228
[126] W. E. Pruitt and S. J. Taylor. Sample path properties of processes with stable components. Z wahrsch verw Gebiete, 1969, 12: 267-289.
[127] W. J. Hendricks. Multiple points for a process in R~2 with stable components, z. w., 1974, 28: 113-128.
[128] W. Kiu. Semi-stable Markov processes in R~n. Stochastic process. Appl., 1980, 10: 205-225.
[129] Wu Liangsen, et al. Mathematical Analysis. Beijing: Higher Education Dress, 2001.
[130] 徐赐文．N指标d维广义Wiener过程象集代数和的性质．数学杂志，1997，17(2)：199-206．
[131] X. Saint Raymond and C. Tricot. Packing regularity of sets in n-space. Math Proc Camb Phil Soc, 1988, 103: 133-145.
[132] Xiong Wei. Hausdorff dimension theorems for processes with self-similar components. 数学杂志,1999, 19: 209-214.
[133] Xu Ciwen. The Hausdorff Measure of Image sets and graph sets of N-parameter d-dimensional generalized Brownian sheet. Act Mathematica Sinica, 2001, 44(6): 1129-1142.
[134] 肖益民．二参数Ornstein-Unlenbeck过程的自相交局部时与重点．待发表．
[135] 肖益民．某些Gauss向量场的几何性质．武汉大学博士学位论文，1990．
[136] 肖益民．某些高斯场的极性与逆象．数学杂志．1991，11(1)：101-108．
[137] Xiao Yimin. Hitting probabilities for Fractional Brownian Motion. Stochastics and stochastics Reports, 1999, 66: 121-151.
[138] Xiao Yimin. Holder condintions for the local times and the Hausderff measure of the level sets of Garssian Kandom fields. Prob Theory Related Fields, 1997, 109: 129-157.
[139] Xiao Yimin. Packing measure of the sample paths of fractional Brownian motion. Tran Amer Math Soc, 1996, 348(8): 3193-3213.
[140] Xiao Yimin. Packing dimension, Hausdorff dimension and Cartesian product sets. Math. Proc. Camb. Phil. Soc., 1996, 120: 535-546.
[141] Xiao Yimin. Hausdorff-type measure of the sample paths of fractional Brownian motion. Stoch Proc Appl, 1998, 74: 251-272.
[142] Xiao Yimin. Hansdorff measure of the sample paths of Gaussian random fields. Osaka J Math, 1996, 33: 895-913.
[143] Xiao Yimin and Liu Luqin. Packing dimension results for certain Lévy processes and self-similar Markov rocesses. Chinese Ann. Math., 1996, 7(A): 392-400.
[144] Xiao Yimin. Dimension results for Gaussian vector fields and index-α stable fields. Ann. Prob., 1995, 23: 273-291.
[145] Xiao Yimin. Packing dimension and fractal properties of stochastic processes. A Lecture in Utal University. 2001. 4.
[146] 杨新建．扩散过程样本的Hlder连续性及其应用．湖南师范大学学报，1995，18(2)：13～18．
[147] 杨新建．扩散过程的极性的一个充分条件．数理统计与应用概率，1996，11(4)：280～290．
[148] 杨新建．非退化扩散过程的极性的必要条件．系统科学与数学，2002，22(1)：67～77．
[149] Z. Ciesielski and S. J. Taylor. First passage times and sojourn times for Brownian motion in space and the exact Hansdorff measure of the sample path. Trans. Amer. Math. Soc., 1962, 103: 434-450
[150] Zhang Runchum. The Markovian of generalized Brounian sheet and generalized O U P_2. Science in China, Series A, 1985, 28(5): 389-398.
[151] Zhuang Xingwu, Li Xinchun. On the continuity of sample functions of generalized Brownian sheets. Chinese Journal of Applied Probability and statistics, 1987, 3(3): 270-273.
[152] 郑水草．N指标d维广义α-stable过程的局部时[J]．福建师范大学学报，1999，15(2)：16-20．
[153] 郑水草．庄兴无．广义α-stable过程局部时增量的Hlder律．数学物理学报，1999，19(4)：415-423．
[154] 郑水草、胡迪鹤．N指标d维广义α-stable过程的逗留时．武汉大学学报，2002，48(3)：434-450．
[155] 郑水草．广义α-stable过程的确切Hausdorff测度函数．数学物理学报(待发表)
[156] 周先银．平面运动的重点．数学年刊，1992，12A
[157] Zhou xianyin. Hausdorff measure of the level sets of multi-parameter Wiener processes in one dimension. Acta Mathematica Sinica. New Series. 1993. 9(4): 390-400.
[158] 钟玉泉，肖益民．stable单的相交局部时及重点[J]．数学物理学报，1995，15：141-152．
[2] A. Dvorezky, P. Erdos and S. Kakutani. Multiple points of paths of Brownian motion in the plane. Bull. Res Council Isr. Sect., 1954, F3: 364-371.
[3] A. Dvorezky, P. Erdos, S. Kakutani and S. J. Taylor. Points of multiplicity of plane Brownian paths. Bull. Res. Council Isr. Sect., 1958, F7: 175-180.
[4] A. Dvorezky. P. Erdos, S. Kakutani and S. J. Taylor. Triple points of Brownian motion in 3-space. Prec. Comb. Phil. Soc.. 1957. 53: 856-862.
[5] A. Goldman. Points multiples des trajectories des processus gaussiens, Z. W.. 1981, 59: 481-494.
[6] A. Goldman. Mouvement Brownien à plusieurs paramètres: mesure de Hausdorff des trajectoires. Astérisque. 1988.
[7] B. B. Mandelbrot and J. W. Van, Ness. Fractional Brownian motions fracfioral noises and applications. SIAM Review. 1968. 10: 422-437.
[8] B. B. Mandelbrot. Stochastic models for the earth's relif, the shape and fractal dimension of the coastlines and the number-area rule for islands. Proc. Nat. Acad. Sci. USA. 1975. 72: 3825-3828.
[9] B. B. Mandelbrot. On the geometry of hornogeneons turbulence with streas on fractional dimension of the isv-surface of scalars. J. Fluid. Mec.. 1975. 72: 401-416.
[10] B. B. Mandelbrot. Fractals:from. chance and dimension. Freeman. San Francisco. 1977.
[11] B. B. Mandelbrot. The Fractal Geomentry of Nature. Freeman. San Francisco, 1982.
[12] B. E. Fristedt and S. J. Taylor. The packing measure of a aeneral subordinator. Prob Th Rel Fields. 1992. 92: 493-510.
[13] C. A. Rogers. Hausdorff measures. Cambridge University Press. 1970.
[14] C. Bishop and Y. Peres. Packing dimension and Cartesian Products. Trans. Amer. Math. Soc.. 1996. 345: 4433-4445.
[15] C. Tricot. Two definitions of fractional dimension. Math. Proc. Canbridge Phil. Soc. 1982. 91: 57-74.
[16] Chen Zhenlong. Liu Sanyang. Dimension of polar sets for Brownian Sheet. Acta Mathematica Scientia (Series B). 2003. 23B (4): 549-561.
[17] 陈振龙，徐赐文．N指标d维广义Wiener过程像集的一致维数．应用数学学报，2003．26(2)：345-358．
[18] Chen Zhenlong. Polar functions for N-paramenter d-dimensional generalized Brownian sheet. Acta. Mathematica Scientia. 2001. 21A(4): 472-479.
[19] 陈振龙，刘三阳．N指标d维广义Wiener过程象集的m项代数和的几个性质．数学物理学报，2002．22(2)：250-257．
[20] 陈振龙，刘禄勤．d维平稳高斯过程极集的必要条件．武汉大学学报，2002，48(1)：11-14．
[21] 陈振龙，刘禄勤．d维平稳高斯过程的极函数及其相关维数．武汉大学学报2002．48(3)：485-489．
[22] 陈振龙，刘三阳．广义Wiener过程极性的两个充分条件．数学杂志，2003，23(1)：102-107．
[23] 陈振龙，刘三阳．广义α-stable过程的自相交局部时．西安电子科技大学学报，2004，31(2)：310-316．
[24] 陈振龙，刘三阳．广义α-stable过程自相交局部时增量的Hlder律及重点．数学物理学报(待发表)
[25] Chen Zhenlong. The properties of the polar set for the Brownian Sheet. J. Math. (PRC). 17(1997): 373-378.
[26] D. German and J. Horowitz. Occuption densities. Ann. Probab, 1980, 8: 1-67.
[27] D. German, J. Horowitz and Jay Rosn. A local time analysis of intersections of Brownian path in the plane. Ann. Probab., 1984, 12: 86-107.
[28] D. Khoshnevisan. Brownian sheet images and Bessel-Riesz capacity. Trans Amer Math Soc, 1999, 351(7): 2607-2622.
[29] D. Khoshnevisan and Z. Shi. Brownian sheet and capacity. Ann Probab, 1999, 27(3): 1135-1159.
[30] D. Khoshnevisan anti Y. Xiao. Level sets of additive Lévy processes. Ann Probab, 2002, 30(1): 62-100.
[31] D. Khoshnevisan. Some polar sets for the Brownian sheet. Sém. de Prob. ⅩⅩⅪ, Lecture Notes in Mathematics 1655. 190-197. 1997.
[32] D. Khoshnevisan. Multi-Parameter Processes. (To publish)
[33] D. J. Mason and Yimin Xiao. Sample path properties of operator self-similar Gaussian random fields. Ter Veroyatnost i Primenen, 2001. 46: 94-116.
[34] D. Ray. Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion. Trans. Amer. Math. Soc.. 1964. 106: 436-444
[35] D. Mourad and D. L. Pitt. Local nondeterminism and Hausdorff dimension. Pro. Prob. stat.. Seminar on stachastic processes, Boston: Birkhauser, 1986.
[36] F. Testard. Quelques propriétés géométriques de certains processus gaussiens. C. R. Acad. Sc. Paris, 300. Série Ⅰ (1985), 497-500.
[37] F. Testard. Dimension asymétrique et ensembles doublement non polairs. C. R. Acad. Sc. Paris, 303, Série Ⅰ (1986). 579-581.
[38] F. Testard. Polarité. Points multiples et géométrie de certain processus gaussiens, Publ. du Laboratoire de Statistique et Probabilités de l' U. P. S. Toulouse. mars 1986, 1-86.
[39] F. Testard. Polarité. Points Multiples et Gémotrie de Certains Processus Gaussiens. Thése Doctorat soutnue à Orsay. 1987.
[40] F. Rezakhanlou and S. J. Taylor. The packing measure of the graph of a stacess. Astérisque 157-158, 1988: 341-361
[41] G. Samorodnitsky and M. S. taqqu. Stable Non-Gaussian Random Processes. New York: Chapman and Hall. 1994.
[42] H. Kesten. An iterated logarithm law for local time. Duke Math. J, 1965, 32: 447-456.
[43] H. Kesten. Hitting probabilities of single points for processes with stationary independent increments. Trans. Amer. Math. Soc., 1969, 93.
[44] Hu X. Some fractal sets determined by stable processes. Prob. Th. Rel. Fields, 1994, 100: 205-225.
[45] 胡迪鹤等．随机分形引论．武汉：武汉大学出版社，1995
[46] 胡迪鹤，刘禄勤，肖益民，等．随机分形．数学进展，1995，24(3)：193-214．
[47] 胡迪鹤．随机过程论-基础、理论、应用．武汉大学出版社，2000．
[48] 何声武，汪嘉岗，严加安．半鞅与随机分析．北京：科学出版社，1995．
[49] J. Cuzick and J. P. Dupreez. Joint continuity of Gaussian local time. Ann. Probab., 1982, 10, 810-817.
[50] J. Lamperti. Semi-stable stochastic processes. Trans. Amer. Math. Soc., 1962, 104: 62-78.
[51] J. F. LeGall and S. T. Taylor. The packing measure of planar Brownian motion. Progress in Probability and Staistics. Seminar on Stochastic Processec. Boston: Birkhauser, 1986. 139-147.
[5
[52] J. F. LeGall. The exact Hausdorff measure of Browniaa multiple points. Seminar on Stochastic Processes. Boston: Birkhauser, 1986.
[53] J. F. LeGall and S. J. Taylor. The Packing measure of planar Brownian motion. Progress in Probability and Statistics, Seminar on Stochastic Processes, Boston: Birkhauser, 1986.
[54] J. F. LeGall. Sur les function polaires pour les mouvement browien. Seminaire de Probabilites XXII LNM, 1321: 186-189.
[55] J. F. LeGall. The exact Hausdorff measure of Borwnian multiple points Ⅱ. Seminar on stochastic processes. Boston: Birkhauser, 1989.
[56] J. F. LeGall, J. S. Rosen and N. R. Shieh. Muliple points of Levy processes. Ann. Probab, 1989. 17: 503-515.
[57] J. Hawkes. Some dimension theorems for the sample functions of stable processes. Indiana Univ Math J, 1971, 20: 733-738
[58] J. Hawkes. W. E. Pruitt. Uniform dimension results for processes with independent increments. Z. Wahrsch. Verw. Geb., 1974, 28: 277-288.
[59] J. Hawkes. Measures of Hausdorff type and stable processes. Mathematika, 1978, 25: 202-212.
[60] J. Hawkes. Multiple points for symmetric Lévy processes. Math. Proc. Camb. Phil. Soc., 1978. 83: 83-90.
[61] J. Nolan. Local nondeterminism and local times for stable processes. Prob. Th. Rel. Field., 1989, 82: 386-410.
[62] J. P. Kahane. Some Random Series of Functions London. Cambridge University Press, 2nd edition. 1985.
[63] J. P. Kahane. Points multiples des processus de lévy symétriques restreints à un ensemble de valurs du temps. Sém. Anal. Harm. 1981-1982, Publ. Math. Orsay 1983, 83(2): 74-105.
[64] J. P. Kahane. Ensembles aléatoires et dimeninons. Cours au Seminaire de l'Escurial, 1983.
[65] J. Rosen. Self-intersections of random fields. Ann. Prob., 1984, 12: 108-119.
[66] J. Vuolle-Apiala and E. S. Graversen. Duality theory for self-similar Markov processes. Ann. Inst. H. Poincare Probab. Statist., 1988, 22: 376-392.
[67] K. J. Falconer, Fractal Geometry-Mathematical Foundations and Applications. Wiley & Sons, 1990.
[68] 乐成雄．Brown单的一致Packing测度及重点的维数．武汉大学硕士论文，1990．
[69] L. D. Pitt and L. T. Tran. Local sample Path properties of Gaussian fields. Ann Probab, 1979, 7: 477-493.
[70] L. D. Pitt. Local times for Gaussian vector fields. Indiana Univ Math J, 1978, 27: 309-330
[71] 林火南，庄兴无．N指标d维广义Wiener过程的点常返性．数学年刊，1992，13A(3)：344-352．
[72] 林火南．多指标稳定过程的一致维数结果．中国科学(A辑)，1999，29(5)：414-425
[73] L. L. Helms. Introduction to Potential Theory. Wiley, New York. 1969.
[74] Liu luqin and Yimin Xiao. Hausdorff dimension theorems for self-similar Markov processes. Prob. Math. Statis., 1998, 18: 369-393.
[75] Liu luqin. The Hausdorff dimension of the image, graph, and level of self-similar process. Stochastic and Quantum Mechanics, World Scientific press, 1992, 178-188.
[76] L. T. Tran. The Hausdorff dimension of the range of the N-parameter Wiener process. Ann. Prob. 1977, 5: 235-242.
[77] 乐成雄．Brown单的一致Packing测度及重点的维数．武汉大学硕士学位论文，1990．
[78] M. B. Marcus. Hder conditions for Gaussian processes with stationary increments. Trans Amer Math Soc, 1968, 134: 29-52.
[79] M. D. Donsker and S. R. S. Varadhan. On laws of the iterated lagarithm and local times. Comm. Pure Appel. Math., 1977, 30: 707-753.
[80] M. F. Vares. Local times for two-parameter Lévy processes. Stoc. Process. Appl., 1983, 15, 59-82.
[81] M. Ledoux and M. Talagrand. Probabilities in Banach Spaces. Springer Verlag, 1991.
[82] M. Talagrand and Yimin Xiao, Fracional Brownian motion and Packing dimension. J. Theort. Probab. 9(1996), 579-593.
[83] M. Talagrand. Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann Probab, 1995, 23: 767-775.
[84] M. Talagrarrd. Lower classes for fractional Brownian motion. J Theoret Probab, 1996. 9: 191-213.
[85] M. T. Barlow and J. Hawkes. Application de l'entropic métrique a la continuity des temps locaux des processus de Lévy. C. R. Acad. Sci. Paris s é r. I Math., 1985, 301: 237-239.
[86] M. T. Barlow. Necessary and sufficient conditions for the continuity of local time of Lévy processes. Ann. Probab., 1988, 16: 1389-1427.
[87] M. T. Lacey. Limit law for local times of the Brownian sheet. Prob. Th. Rel. Fields. 1990. 86: 63-85.
[88] N. C. Jain and S. E. Pruitt. The correct measure function for the graph of a transient stable process. Z wahrsch verw Gebiete. 1968. 9: 131-138
[89] N. Kno. Sur la minoration asymptotique et le cara ctère transitoire des trajectoires des fonctions aleatories gaussiennes à valeurs dans k~d. Z wahrsch verw Gebiete, 1975. 33: 95-112.
[90] N. Ikeda and S. Watanable. Stochastic Differential Equations and Diffusion Processes. New York: North-Holland, Publishing Company. 1981.
[91] N. N. Cēntsov. Wiener random fields depending several parameters. Dokl Akad Nauk S S S R (NS), 1956, 106: 607-609.
[92] P. Imkeller. Two-parameter Martingales and their quadratic variation. Lecture Notes in Mathematics, 1308, Springer, New York, 1988.
[93] P. Lévy. La mesure de Hausdorff de la courbe du mouvement brownien. Giorn Ital Attuari. 1953, 16: 1-37.
[94] P. Lévy. Processus Stochastiques et Mouvements Browniens. Gauthier-Villars, 1948.
[95] R. J. Adler. Hausdorff dimension and Gaussian fields. Ann. Probab., 1977, 5: 145-151.
[96] R. J. Adler. The uniform dimension of the level sets of a Brownian sheet. Ann. Probab., 1978, 6: 505-515.
[97] R. J. Adler. The Geomentry of Random Fields. New York: Wiley, 1981.
[98] R. Kaufman. Measures of Hausdorff type and Brownian motion. Mathematika. 1972, 19: 115-119.
[99] R. Kaufman. Une propiété métriqé du mouvement brownien. C. R. Acad. Sci. Paris. Sér A, 1969, 268: 727-728.
[100] R. M. Blumenthal and R. K. Getoor. local times for Markor processes. Z. W., 1964, 3: 50-74.
[101] S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory. Academic Press, New York, 1978.
[102] S. Orey and N. E. Pruitt. Sample function of the N parameter Wiener processes. Ann. Prob., 1973, 1: 138-163.
[103] S. E. Graversen and J. Vuolle-Apiala. α-self-similar Markowv processes. Probab. Th. Rel. Fidlds, 1986, 71: 149-158.
[104] S. E. Graversen." Polar-functions " for Brownian motion. Z. Wahrsch. Verw Gebiete. 1982,61:261-270.
[105] S. J. Taylor. The Hausdorff Q-dimensional measure of Brownian paths in n-space. Proc. Cambridge Philos. Soc, 1953, 49:31-39.
[106] S. J. Taylor. The dimensional measure of the graph and set of zeros of a Brownian path. Proc. Cambridge Philos. Soc.,1955, 51:265-274.
[107] S. J. Taylor. The exact Hausdorff measure of the sample path for planar Brownian motion. Proc. Cambridge Phil. Soc, 1964,60:253-258.
[108] S. J. Taylor. Multiple points for the sample paths of the symmetric processes. Z. W., 1966, 5: 247-264.
[109] S. J. Taylor. The measure theory of random fractal. Math. Proc. Cambridge Phil. Soc, 1986,100(3):383-406.
[110] S. J. Taylor and N. A. Watson. A Hausdorff measure classification of polar sets for the heat equation. Math.Proc.Camb. Philos. Soc, 1985. 97: 325-344.
[111] S. J. Taylor. The use of Packing measure in the analysis of random sets. Lecture Notes in Math. 1986, 1203:214-222.
[112] S. J. Taylor and C. Tricot. Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc, 1985, 288: 679-699.
[113] S. J. Sheu. Some estimates of the transition density of a nondegener ate diffusion Markov process.Ann.Probab..1991,19(2):538-561.
[114] S. Kakutani. Two-dimensional Brownian motion and harmonic functions. Proc. Imperial Acad. Tokyo 20(1944), 706-714.
[115] S. M. Berman. Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc, 1969, 137: 277-300.
[116] S. M. Berman. Gaussian Processes with stationary increments, local times and sample function properties. Ann. Statist.,1970, 41: 1260-1272.
[117] S. M. Berman. Gaussian sample functions: uniform dimension and Holder condition nowhere. Nagoya Math. J., 1972, 46: 63-88.
[118] S. M. Berman. Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J., 1973, 23: 69-94.
[119] S. Song. Inegalites relatives aux procesus d'ornstein-Uhlenbeck a n-parametres et capacite gausienne Cn,2. Seminare de Probabilities XXVII. Lecture Notes in Math. 1557:276-301. Spring, Berlin.
[120] Theory of the Integral. New York: Dover, 1964.
[121] T. S. Mountford. An extension of a result of Kahane using Brownian local times of intersections. Stochastics, 1988,23:449-464.
[122] T. S. Mountford. A relation between Hausdorff dimension and a condition on time sets for the image by the Brownian Sheet to possess interior points. Bull Lon. Math. Soc. 1989, 21: 179-185.
[123] T. S. Mountford. Uniform dimension results for the Brownian Sheet. Ann. Prob., 1989, 4:1454-1462.
[124] T. S. Salisbury. Energy.and intersection of Markov chains. In Random Discrete Structures 213- 225. New York: Springer,1995.
[125] W. Ehm. Sample function properties of multi-parameter stable, processes. Z Wahrsch Verw Geb, 1981, 56, 195-228
[126] W. E. Pruitt and S. J. Taylor. Sample path properties of processes with stable components. Z wahrsch verw Gebiete, 1969, 12: 267-289.
[127] W. J. Hendricks. Multiple points for a process in R~2 with stable components, z. w., 1974, 28: 113-128.
[128] W. Kiu. Semi-stable Markov processes in R~n. Stochastic process. Appl., 1980, 10: 205-225.
[129] Wu Liangsen, et al. Mathematical Analysis. Beijing: Higher Education Dress, 2001.
[130] 徐赐文．N指标d维广义Wiener过程象集代数和的性质．数学杂志，1997，17(2)：199-206．
[131] X. Saint Raymond and C. Tricot. Packing regularity of sets in n-space. Math Proc Camb Phil Soc, 1988, 103: 133-145.
[132] Xiong Wei. Hausdorff dimension theorems for processes with self-similar components. 数学杂志,1999, 19: 209-214.
[133] Xu Ciwen. The Hausdorff Measure of Image sets and graph sets of N-parameter d-dimensional generalized Brownian sheet. Act Mathematica Sinica, 2001, 44(6): 1129-1142.
[134] 肖益民．二参数Ornstein-Unlenbeck过程的自相交局部时与重点．待发表．
[135] 肖益民．某些Gauss向量场的几何性质．武汉大学博士学位论文，1990．
[136] 肖益民．某些高斯场的极性与逆象．数学杂志．1991，11(1)：101-108．
[137] Xiao Yimin. Hitting probabilities for Fractional Brownian Motion. Stochastics and stochastics Reports, 1999, 66: 121-151.
[138] Xiao Yimin. Holder condintions for the local times and the Hausderff measure of the level sets of Garssian Kandom fields. Prob Theory Related Fields, 1997, 109: 129-157.
[139] Xiao Yimin. Packing measure of the sample paths of fractional Brownian motion. Tran Amer Math Soc, 1996, 348(8): 3193-3213.
[140] Xiao Yimin. Packing dimension, Hausdorff dimension and Cartesian product sets. Math. Proc. Camb. Phil. Soc., 1996, 120: 535-546.
[141] Xiao Yimin. Hausdorff-type measure of the sample paths of fractional Brownian motion. Stoch Proc Appl, 1998, 74: 251-272.
[142] Xiao Yimin. Hansdorff measure of the sample paths of Gaussian random fields. Osaka J Math, 1996, 33: 895-913.
[143] Xiao Yimin and Liu Luqin. Packing dimension results for certain Lévy processes and self-similar Markov rocesses. Chinese Ann. Math., 1996, 7(A): 392-400.
[144] Xiao Yimin. Dimension results for Gaussian vector fields and index-α stable fields. Ann. Prob., 1995, 23: 273-291.
[145] Xiao Yimin. Packing dimension and fractal properties of stochastic processes. A Lecture in Utal University. 2001. 4.
[146] 杨新建．扩散过程样本的Hlder连续性及其应用．湖南师范大学学报，1995，18(2)：13～18．
[147] 杨新建．扩散过程的极性的一个充分条件．数理统计与应用概率，1996，11(4)：280～290．
[148] 杨新建．非退化扩散过程的极性的必要条件．系统科学与数学，2002，22(1)：67～77．
[149] Z. Ciesielski and S. J. Taylor. First passage times and sojourn times for Brownian motion in space and the exact Hansdorff measure of the sample path. Trans. Amer. Math. Soc., 1962, 103: 434-450
[150] Zhang Runchum. The Markovian of generalized Brounian sheet and generalized O U P_2. Science in China, Series A, 1985, 28(5): 389-398.
[151] Zhuang Xingwu, Li Xinchun. On the continuity of sample functions of generalized Brownian sheets. Chinese Journal of Applied Probability and statistics, 1987, 3(3): 270-273.
[152] 郑水草．N指标d维广义α-stable过程的局部时[J]．福建师范大学学报，1999，15(2)：16-20．
[153] 郑水草．庄兴无．广义α-stable过程局部时增量的Hlder律．数学物理学报，1999，19(4)：415-423．
[154] 郑水草、胡迪鹤．N指标d维广义α-stable过程的逗留时．武汉大学学报，2002，48(3)：434-450．
[155] 郑水草．广义α-stable过程的确切Hausdorff测度函数．数学物理学报(待发表)
[156] 周先银．平面运动的重点．数学年刊，1992，12A
[157] Zhou xianyin. Hausdorff measure of the level sets of multi-parameter Wiener processes in one dimension. Acta Mathematica Sinica. New Series. 1993. 9(4): 390-400.
[158] 钟玉泉，肖益民．stable单的相交局部时及重点[J]．数学物理学报，1995，15：141-152．