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Local times and related properties of multidimensional iterated Brownian motion. (English) Zbl 0914.60063
Let {W(t),tR} and {B(t),t0} 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)=(X 1 (t),,X d (t)) where X 1 ,,X d are independent copies of Y. The author investigates the existence, joint continuity and Hölder conditions in the set variable of the local time L={L(x,B):xR d , B(R + )} of X(t), where ( + ) is the Borel σ-algebra of R + . Finally 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).

MSC:
60J65Brownian motion