The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion (Q2372571)

Let \((W(t),t\in {\mathbb R})\), \((B(t),t\in {\mathbb R_{+}})\) be two independent Brownian motions on \({\mathbb R}\) with \(W(0)=B(0)=0\). The iterated Brownian motion is defined by \(Y(t)=W(B(t))\), \(t\in {\mathbb R_{+}}\). \textit{Y. Xiao} [J. Theor. Probab. 11, No. 2, 383--408 (1998; Zbl 0914.60063)] introduced a \(d\)-dimensional iterated Brownian motion \(X(t)=(X_1(t),\dots,X_d(t))\) where \(X_1(t),\dots,X_d(t)\) are independent copies of \(Y\) and obtained a lower bound of the Hausdorff measure of the graph set as \(d<4\). The method of Xiao cannot be used for \(d>4\). In this paper the authors use a different tool -- the sojourn time -- to solve this problem. For a related paper see: \textit{Y. Xiao} [Probab. Theory Relat. Fields 109, No. 1, 129--157 (1997; Zbl 0882.60035)].