Polar Functions for Anisotropic Gaussian Random Fields
Let X be an (N, d)-anisotropic Gaussian random field. Under some general conditions on X, we establish a relationship between a class of continuous functions satisfying the Lipschitz condition and a class of polar functions of X. We prove upper and lower bounds for the intersection probability for a...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/947171 |
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| _version_ | 1832564470153478144 |
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| author | Zhenlong Chen |
| author_facet | Zhenlong Chen |
| author_sort | Zhenlong Chen |
| collection | DOAJ |
| description | Let X be an (N, d)-anisotropic Gaussian random field. Under some general conditions on X, we establish a relationship between a class of continuous functions satisfying the Lipschitz condition and a class of polar functions of X. We prove upper and lower bounds for the intersection probability for a nonpolar function and X in terms of Hausdorff measure and capacity, respectively. We also determine the Hausdorff and packing dimensions of the times set for a nonpolar function intersecting X. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise, and the operator-scaling Gaussian random field with stationary increments. |
| format | Article |
| id | doaj-art-55e2fbe022b04206913199cc1c646741 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-55e2fbe022b04206913199cc1c6467412025-02-03T01:10:51ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/947171947171Polar Functions for Anisotropic Gaussian Random FieldsZhenlong Chen0School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, ChinaLet X be an (N, d)-anisotropic Gaussian random field. Under some general conditions on X, we establish a relationship between a class of continuous functions satisfying the Lipschitz condition and a class of polar functions of X. We prove upper and lower bounds for the intersection probability for a nonpolar function and X in terms of Hausdorff measure and capacity, respectively. We also determine the Hausdorff and packing dimensions of the times set for a nonpolar function intersecting X. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise, and the operator-scaling Gaussian random field with stationary increments.http://dx.doi.org/10.1155/2014/947171 |
| spellingShingle | Zhenlong Chen Polar Functions for Anisotropic Gaussian Random Fields Abstract and Applied Analysis |
| title | Polar Functions for Anisotropic Gaussian Random Fields |
| title_full | Polar Functions for Anisotropic Gaussian Random Fields |
| title_fullStr | Polar Functions for Anisotropic Gaussian Random Fields |
| title_full_unstemmed | Polar Functions for Anisotropic Gaussian Random Fields |
| title_short | Polar Functions for Anisotropic Gaussian Random Fields |
| title_sort | polar functions for anisotropic gaussian random fields |
| url | http://dx.doi.org/10.1155/2014/947171 |
| work_keys_str_mv | AT zhenlongchen polarfunctionsforanisotropicgaussianrandomfields |