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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 ϕ(x)=x 3/4 [loglog(1/x)] 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:
60J65Brownian motion
28A78Hausdorff and packing measures
60J55Local time, additive functionals