| The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion |
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| 作者姓名： | Rong-mao ZHANG & Zheng-yan LIN |
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| 作者单位： | Rong-mao ZHANG & Zheng-yan LIN Department of Mathematics，Zhejiang University，Hangzhou 310027，China |
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| 基金项目： | 国家自然科学基金;中国博士后科学基金 |
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| 摘 要： | Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.
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| 收稿时间： | 2 November 2005 |
| 修稿时间： | 22 June 2006 |
The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion |
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| Rong-mao ZHANG & Zheng-yan LIN.The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion[J].Science in China(Mathematics),2007,50(1):35-46. |
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| Authors: | Rong-mao ZHANG Zheng-yan LIN |
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| Institution: | Department of Mathematics, Zhejiang University, Hangzhou 310027, China |
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| Abstract: | Let {W (t), t ∈ R}, {B(t), t ∈ R
+} be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (X
1(t),…, X
d
(t)) and X
1(t),…, X
d
(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q ⊆ (0, ∞), the exact Hausdorff measures of the image X(Q) = {X(t): t ∈ Q} and the graph GrX(Q) = {(t, X(t)): t ∈ Q} are established.
This work was partially supported by the National Natural Science Foundation of China (Grant No. 10131040), China Postdoctoral
Science Foundation. |
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| Keywords: | graph Hausdorff measure image iterated Brownian motion sojourn time |
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