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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 $Z(t)=X(Y(t))$ 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 $\varphi(x)=x^{3/4}[\log\log(1/x)]^{3/4}$. This result generalizes earlier work of {\it K. Burdzy} and {\it 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.

60J65Brownian motion
28A78Hausdorff and packing measures
60J55Local time, additive functionals
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