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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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