Local Times and Related Properties of Multidimensional Iterated Brownian Motion

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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)) (t ≥ 0) be the iterated Brownian motion. Define d-dimensional iterated Brownian motion by X(t)=(X1(t),…,}Xd(t)) (t ≥ 0) where X1,…, Xd 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 ∈ Rd, B ∈ ℛ(R+)} of X(t), where ℬ(B+) 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).

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