Abstract
Let \( \tilde W \) (t)(t ∈ R N+ ) be the d-dimensional N-parameter generalized Brownian sheet. We study the polar sets for \( \tilde W \) (t). It is proved that for any a ∈ R d, P{\( \tilde W \) (t) = a, for some t ∈ R N> } = \( \left\{ {\begin{array}{*{20}c} {1,{\mathbf{ }}if{\mathbf{ }}\beta d < 2N} \\ {0,{\mathbf{ }}if{\mathbf{ }}\alpha d > 2N} \\ \end{array} } \right. \) and the probability that \( \tilde W \) (t) has k-multiple points is 1 or 0 according as whether 2kN > d(k − 1)β or 2kN < d(k−1)α. These results contain and extend the results of the Brownian sheet, where R N> = (0,+∞)N,R N+ =[10,+∞)N,0< α≤1 and β≥1.
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Foundation item: Supported by the National Natural Science Foundation of China (10471148) and the Natural Science Foundation of Shaanxi Province (2005A08, 2006A14)
Biography: LI Huiqiong (1966–), female, Associate professor, research direction: stochastic process and random fractal.
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Li, H., Liu, L. & Chen, Z. Some polar sets for the generalized Brownian sheet. Wuhan Univ. J. Nat. Sci. 13, 137–140 (2008). https://doi.org/10.1007/s11859-008-0203-4
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DOI: https://doi.org/10.1007/s11859-008-0203-4