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Hausdorff and packing measures of the level sets of iterated Brownian motion. (English) Zbl 0935.60066
Iterated Brownian motion is defined as $Z\left(t\right)=X\left(Y\left(t\right)\right)$ where $X$ and $Y$ are independent (two-sided) Brownian motions. The exact Hausdorff dimension gauge of the level sets of iterated Bronwian motion is determined to be $\varphi \left(x\right)={x}^{3/4}{\left[loglog\left(1/x\right)\right]}^{3/4}$. This result generalizes earlier work of K. Burdzy and D. Khoshnevisan [in: Séminaire de probabilités XXIX. Lect. Notes Math. 1613, 231-236 (1995; Zbl 0853.60061)]. The paper also contains a slightly less precise result about the packing gauge of the level sets. The proofs rely on an accurate analysis of the local times.
MSC:
 60J65 Brownian motion 28A78 Hausdorff and packing measures 60J55 Local time, additive functionals