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Local Times and Related Properties of Multidimensional Iterated Brownian Motion

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  • Yimin Xiao

Abstract

Let {W(t), t∈R} and {B(t), t≥0} be two independent Brownian motions in R with W(0) = B(0) = 0 and let $$Y(t) = W(B(t)){\text{ }}(t \geqslant 0)$$ be the iterated Brownian motion. Define d-dimensional iterated Brownian motion by $$X(t) = (X_1 (t),...,X_d (t)){\text{ }}(t \geqslant 0)$$ where X 1, X d are independent copies of Y. In this paper, we investigate the existence, joint continuity and Hölder conditions in the set variable of the local time $$L = \{ L(x,B):x \in {\text{R}}^d ,B \in B({\text{R}}_{\text{ + }} )\}$$ of X(t), where $$B({\text{R}}_{\text{ + }} )$$ is the Borel σ-algebra of R +. These results are applied to study the irregularities of the sample paths and the uniform Hausdorff dimension of the image and inverse images of X(t).

Suggested Citation

  • Yimin Xiao, 1998. "Local Times and Related Properties of Multidimensional Iterated Brownian Motion," Journal of Theoretical Probability, Springer, vol. 11(2), pages 383-408, April.
    • Handle: RePEc:spr:jotpro:v:11:y:1998:i:2:d:10.1023_a:1022679721638
    DOI: 10.1023/A:1022679721638
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    References listed on IDEAS

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    1. Hu, Y. & Shi, Z., 1995. "The Csörgo-Révész modulus of non-differentiability of iterated Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 58(2), pages 267-279, August.
    2. Bertoin, Jean, 1996. "Iterated Brownian motion and stable() subordinator," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 111-114, April.
    3. Shi, Z., 1995. "Lower limits of iterated Wiener processes," Statistics & Probability Letters, Elsevier, vol. 23(3), pages 259-270, May.
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    Cited by:

    1. Yueyun Hu, 1999. "Hausdorff and Packing Measures of the Level Sets of Iterated Brownian Motion," Journal of Theoretical Probability, Springer, vol. 12(2), pages 313-346, April.

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