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The level sets of iterated Brownian motion. (English) Zbl 0853.60061
Azéma, J. (ed.) et al., Séminaire de probabilités XXIX. Berlin: Springer-Verlag. Lect. Notes Math. 1613, 231-236 (1995).
An iterated Brownian motion \(Z= (Z(t): t\geq 0)\) is a process defined by \(Z(t)= X(Y (t))\), where \(X= (X(s): -\infty< s< \infty)\) is a one-dimensional Brownian motion indexed by the whole real line, and \(Y= (Y(t): t\geq 0)\) a standard one-dimensional Brownian motion which is independent of \(X\). For every real number \(x\) and \(t\geq 0\), define the level set \({\mathcal L}_x (t)= \{0\leq s\leq t: Z(s)= x\}\). The main result is that almost surely, the Hausdorff dimension of \({\mathcal L}_x (t)\) is \(3/4\), simultaneously for all \(t\geq 0\) and \(x\) in the interior of \(Z([ 0,t])\). The proof relies on the one hand on an upper bound for the modulus of continuity of the local times of \(Z\), and on the other hand on the fact that the paths of an iterated Brownian motion are a.s. Hölder continuous of order \({1\over 4}- \varepsilon\) for any \(\varepsilon> 0\).
For the entire collection see [Zbl 0826.00027].

MSC:
60J65 Brownian motion
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