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Local times and related properties of multidimensional iterated Brownian motion. (English) Zbl 0914.60063
Let \(\{W(t), t\in R\}\) and \(\{B(t), t\geq 0\}\) 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 \(X(t)=(X_1(t),\dots,X_d(t))\) where \(X_1,\dots,X_d\) are independent copies of \(Y\). The author investigates the existence, joint continuity and Hölder conditions in the set variable of the local time \(L=\{L(x,B):x\in R^d\), \(B\in{\mathcal B}(R_+)\}\) of \(X(t)\), where \(({\mathcal B}_+)\) is the Borel \(\sigma\)-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:
60J65 Brownian motion
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