| Hausdorff dimension of set generated by exceptional oscillations of a class of N-parameter Gaussian processes |
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| 作者姓名： | 林正炎 程宗毛 |
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| 作者单位： | [1]Department of Mathematics, Zhejiang University, Hangzhou 310028, P. R. China; [2]Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, P. R. China |
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| 基金项目： | Project supported by the National Natural Science Foundation of China （No.10571159） and the Doctoral Foundation of Ministry of Education of China （No.20060335032）; Acknowledgements The authors are pleased to acknowledge the Hangzhou Dianzi University Foundation （No.KYS091506042） for supporting this research work. |
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| 摘 要： | A class of N-parameter Gaussian processes are introduced, which are more general than the N-parameter Wiener process. The definition of the set generated by exceptional oscillations of a class of these processes is given, and then the Hausdorff dimension of this set is defined. The Hausdorff dimensions of these processes are studied and an exact representative for them is given, which is similar to that for the two-parameter Wiener process by Zacharie （2001）. Moreover, the time set considered is a hyperrectangle which is more general than a hyper-scluare used by Zacharie （2001）. For this more general case, a Fernique-type inequality is established and then using this inequality and the Slepian lemma, a Levy＇s continuity modulus theorem is shown. Independence of increments is required for showing the representative of the Hausdorff dimension by Zacharie （2001）. This property is absent for the processes introduced here, so we have to find a different way.
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| 关 键 词： | 高斯过程 随机过程 豪斯多夫维数 控制论 |
| 收稿时间： | 2006-09-26 |
| 修稿时间： | 2006-11-13 |
Hausdorff dimension of set generated by exceptional oscillations of a class of <Emphasis Type="Italic">N</Emphasis>-parameter Gaussian processes |
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| Lin?Zheng-yan,Cheng?Zong-mao.Hausdorff dimension of set generated by exceptional oscillations of a class of N-parameter Gaussian processes[J].Applied Mathematics and Mechanics(English Edition),2007,28(2):237-245. |
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| Authors: | Lin Zheng-yan Cheng Zong-mao |
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| Institution: | (1) Department of Mathematics, Zhejiang University, Hangzhou, 310028, P. R. China;(2) Department of Mathematics, Hangzhou Dianzi University, Hangzhou, 310018, P. R. China |
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| Abstract: | A class of N-parameter Gaussian processes are introduced, which are more general than the N-parameter Wiener process. The definition of the set generated by exceptional oscillations of a class of these processes is given, and then the Hausdorff dimension of this set is defined. The Hausdorff dimensions of these processes are studied and an exact representative for them is given, which is similar to that for the two-parameter Wiener process by Zacharie (2001). Moreover, the time set considered is a hyperrectangle which is more general than a hyper-square used by Zacharie (2001). For this more general case, a Fernique-type inequality is established and then using this inequality and the ments is required for showing the representative of the Hausdorff dimension by Zacharie (2001). This property is absent for the processes introduced here, so we have to find a different way. |
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| Keywords: | N-parameter Gaussian process modulus of continuityl Hausdorff dimension |
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