Let W^~ be a two-parameter, R^d-valued generalized Brownian sheet. The author obtains an explicit Bessel-Riesz capacity estimate for the images of a two-dimensional set under W^~. He also presents the connections betw...Let W^~ be a two-parameter, R^d-valued generalized Brownian sheet. The author obtains an explicit Bessel-Riesz capacity estimate for the images of a two-dimensional set under W^~. He also presents the connections between the Lebesgue measure of the image of W^~ and Bessel-Riesz capacity. His conclusions also solve a problem proposed by J.- P.Kahane.展开更多
Let W^-（t）（t∈R＋^N） be the d-dimensional N-parameter generalized Brownian sheet. We study the polar sets for W^-（t）. It is proved that for any α∈ R^d, P{W^-（t） = α, for some t∈ R〉^N} = {1, if βd 〈 2N ...Let W^-（t）（t∈R＋^N） be the d-dimensional N-parameter generalized Brownian sheet. We study the polar sets for W^-（t）. It is proved that for any α∈ R^d, P{W^-（t） = α, for some t∈ R〉^N} = {1, if βd 〈 2N ,0 if αd〉 2N and the probability that 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,＋∞）U,R＋^N = [0,＋∞）^N,0〈 α ≤1and β〉1.展开更多
Let W^～=^～{W^～（t）; t∈ R＋^N） be a d-dimensional N-parameter generalized Brownian sheet. Necessary and sufficient conditions for a compact set E × F to be a polar set for （t,W^～（t）） are proved. It is a...Let W^～=^～{W^～（t）; t∈ R＋^N） be a d-dimensional N-parameter generalized Brownian sheet. Necessary and sufficient conditions for a compact set E × F to be a polar set for （t,W^～（t）） are proved. It is also proved that if 2N ≤αd, then for any compact set E ∩→ R〉^N,d-2/α Dim E≤inf{dim F：F∈B（R^d）,P{W^～（E）∩F≠0}〉0}≤d-2/βDimE,and if 2N〉αd, then for any compact set F∪→R^d／{0},α/2（d-DimF）≤inf{dimE：E∈B（R〉^N）,P{W^～（E）∩F≠0}〉0}≤β/2（d-DimF）,where B（R^d） and B（R〉^N） denote the Borel σ-algebra in R^d and in R〉^N respectively, dim and Dim are Hausdorff dimension and Packing dimension respectively.展开更多
For the N-parameter d-dimensional Generalized Brownian Sheet (in short G. B. S.) satisfying some conditions, this paper proves that when 2N≤αd, with probability one, E∈B(R_+~N), there holds 2/βdim E+2N-2Nβ/α≤d...For the N-parameter d-dimensional Generalized Brownian Sheet (in short G. B. S.) satisfying some conditions, this paper proves that when 2N≤αd, with probability one, E∈B(R_+~N), there holds 2/βdim E+2N-2Nβ/α≤dim (E)≤2/αdim E. Especially when α=β=1, we have dim (E)=2 dim E. Noting that even if α=β=1, the G. B. S. is wider than the Brownian Sheet, thus we have extended the uniform dimension result of Mountford, T,S.展开更多
基金Supported by the key research base for humanities and social sciences of Zhejiang Provincial high education talents (Statistics of Zhejiang Gongshang University)
文摘Let W^~ be a two-parameter, R^d-valued generalized Brownian sheet. The author obtains an explicit Bessel-Riesz capacity estimate for the images of a two-dimensional set under W^~. He also presents the connections between the Lebesgue measure of the image of W^~ and Bessel-Riesz capacity. His conclusions also solve a problem proposed by J.- P.Kahane.
基金the National Natural Science Foundation of China (10471148)the Natural Science Foundation of Shaanxi Province (2005A08, 2006A14)
文摘Let W^-（t）（t∈R＋^N） be the d-dimensional N-parameter generalized Brownian sheet. We study the polar sets for W^-（t）. It is proved that for any α∈ R^d, P{W^-（t） = α, for some t∈ R〉^N} = {1, if βd 〈 2N ,0 if αd〉 2N and the probability that 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,＋∞）U,R＋^N = [0,＋∞）^N,0〈 α ≤1and β〉1.
基金Key Research Base for Humanities and Social Sciences of Zhejiang Provincial High Education Talents(Statistics of Zhejiang Gongshang University)
文摘Let W^～=^～{W^～（t）; t∈ R＋^N） be a d-dimensional N-parameter generalized Brownian sheet. Necessary and sufficient conditions for a compact set E × F to be a polar set for （t,W^～（t）） are proved. It is also proved that if 2N ≤αd, then for any compact set E ∩→ R〉^N,d-2/α Dim E≤inf{dim F：F∈B（R^d）,P{W^～（E）∩F≠0}〉0}≤d-2/βDimE,and if 2N〉αd, then for any compact set F∪→R^d／{0},α/2（d-DimF）≤inf{dimE：E∈B（R〉^N）,P{W^～（E）∩F≠0}〉0}≤β/2（d-DimF）,where B（R^d） and B（R〉^N） denote the Borel σ-algebra in R^d and in R〉^N respectively, dim and Dim are Hausdorff dimension and Packing dimension respectively.
基金Supported by the National Natural Science Foundation of China
文摘For the N-parameter d-dimensional Generalized Brownian Sheet (in short G. B. S.) satisfying some conditions, this paper proves that when 2N≤αd, with probability one, E∈B(R_+~N), there holds 2/βdim E+2N-2Nβ/α≤dim (E)≤2/αdim E. Especially when α=β=1, we have dim (E)=2 dim E. Noting that even if α=β=1, the G. B. S. is wider than the Brownian Sheet, thus we have extended the uniform dimension result of Mountford, T,S.
