Local Times and Related Properties of Multidimensional Iterated Brownian Motion

Yimin Xiao

Journal of Theoretical Probability, 1998, vol. 11, issue 2, 383-408

Abstract: 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).

Keywords: Iterated Brownian motion; Local times; Hölder conditions; Level set; Hausdorff dimension (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1023/A:1022679721638

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