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Local Times and Related Properties of Multidimensional Iterated Brownian Motion

Authors:

Yimin Xiao

Abstract:

Let {W(t), t isin

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 sgr

-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 Hö lder conditions Level set Hausdorff dimension

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