The local time of iterated Brownian motion
The study of the iterated Brownian motion \(H=\{H(t): = W_1(|W_2(t) |)\), \(t\geq 0\}\), where \(W_1\) and \(W_2\) are two independent Wiener processes, has motivated numerous works in the recent years. The purpose of the paper is to investigate the local time processes \(L^* = \{L^*(x,t), x\in {\mathbf R}\) and \(t \geq 0\}\) of \(H\), which is formally defined as \(L^*(x,t) = \int^t_0 \delta_x (H(s))ds\), where \(\delta_x\) stands for the Dirac point mass at \(x\). The authors establish a strong approximation result for \(L^*(x,t)\) in terms of the number of visits of a given site by an iterated (simple symmetric) random walk. Further results on the regularity of the local time process are proven, including the so-called upper-upper, upper-lower, lower-upper and lower-lower classes.
scientific article; zbMATH DE number 934236
scientific article