Li, Huiqiong; Liu, Luqin; Chen, Zhenlong Some polar sets for the generalized Brownian sheet. (English) Zbl 1199.60302 Wuhan Univ. J. Nat. Sci. 13, No. 2, 137-140 (2008). Summary: Let \(\widetilde{W}(t)(t\in \mathbf{R}^N_+)\) be the \(d\)-dimensional \(N\)-parameter generalized Brownian sheet. We study the polar sets for \(\widetilde{W}(t)\). It is proved that for any \(a\in \mathbf{R}^d\), \[ P\{\widetilde{W}(t)=a, \text{ for some } t\in \mathbf{R}^N_>\}=\begin{cases} 1, \text{ if } \beta d<2N \\ 0, \text{ if } \alpha d>2N \end{cases} \] and the probability that \(\widetilde{W}(t)\) has \(k\)-multiple points is 1 or 0 according as whether \(2kN>d(k-1)\beta\) or \(2kN<d(k-1)\alpha\). These results contain and extend the results of the Brownian sheet, where \(\mathbf{R}^N_>=(0,+\infty)^N,\mathbf{R}^N_+=[0,+\infty)^N,0<\alpha\leqslant 1\) and \(\beta\geqslant 1\). MSC: 60J65 Brownian motion Keywords:generalized Brownian sheet; polar set; single point; multiple points PDFBibTeX XMLCite \textit{H. Li} et al., Wuhan Univ. J. Nat. Sci. 13, No. 2, 137--140 (2008; Zbl 1199.60302) Full Text: DOI References: [1] Kakutani S. Two-Dimensional Brownian Motion and Harmonic Functions [J]. Proc Imperial Acad, 1944, 20: 706–714. · Zbl 0063.03107 [2] Hawkes J. Measures of Hausdorff Type and Stable Processes [J]. Mathematika, 1978, 25: 202–212. · Zbl 0399.60067 [3] Kahane J P. Points Multiples des Processus de Lévy Symétriques Restreints à Unensemble de Valeurs du Temps [J]. Sém Anal Harm, 1983, 38(2): 74–105. · Zbl 0512.60069 [4] Testard F. Quelques Propriétés Géomé Triques de Certains Processus Gaussiens [J]. C R Acad Sc., 1985, 300: 497–500. · Zbl 0575.60038 [5] Testard F. Polarité, Points, Multiples et Géométrie de Certain Processus Gaussiens [J]. Publ du Laboratoire de Statistique et Probabilités de l’UPS, 1986, 3: 1–86. [6] Xiao Y M. Hitting Probabilities and Polar Sets for Fractional Brownian Motion [J]. Stochastics and Stochastics Reports, 1999, 66: 121–151. · Zbl 0928.60027 [7] Khoshnevisan D. Some Polar Sets for the Brownian Sheet [J]. Lecture Notes in Mathematics, 1997, 1655: 190–197. · Zbl 0886.60039 [8] Chen Zhenglong. The Properties of the Polar Sets for N-Parameter d-Dimensional Generalized Wiener Process [J]. Acta Math Sci, 2000, 20(3): 394–399 (Ch). · Zbl 0961.60047 [9] Taylor S J, Watson N A. A Hausdorff Measure Classification of Polar Sets for the Heat Equation [J]. Math Proc Camb Phil Soc, 1985, 97: 325–344. · Zbl 0584.31006 [10] Zhang Runchu. Markovian Properties of Generalized Brownian Sheet and Generalised OUP2 Process [J]. China Science (A), 1985, 25(5); 389–398. [11] Taylor S J. The Measure Theory of Random Fractals [J]. Math Proc Camb Phil Soc, 1986, 100: 383–406. · Zbl 0622.60021 [12] Xu Ciwen. The Properties of Algebraic Sum of Image Set of N-d Dimensional Generalized Brownian Sheet [J]. J Math, 1997, 17(2): 199–206 (Ch). · Zbl 0933.60038 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.