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Some path properties of iterated Brownian motion. (English) Zbl 0789.60060
Cinlar, E. (ed.) et al., Seminar on stochastic processes, 1992. Held at the Univ. of Washington, DC, USA, March 26-28, 1992. Basel: Birkhäuser. Prog. Probab. 33, 67-87 (1992).
The so-called iterated Brownian motion is the process \(Z=(Z(t), t \geq 0)\) defined by \(Z(t)=X(Y(t))\), where \((X(t), t \geq 0)\), \((X(-t), t \geq 0)\) and \((Y(t), t \geq 0)\) are three independent standard Brownian motions. The first concern of the paper is to decide whether \(Z\) specifies \(X\) and \(Y\). The answer is surprisingly nearly positive, in the sense that one can construct from \(Z\) two paths \((X'(t), -\infty<t<\infty)\) and \((Y'(t), t \geq 0)\) such that \[ X(t)=X'(t) \text{ for all } t \in (-\infty,\infty) \text{ and } Y(t)=Y'(t) \text{ for all } t \in [0,\infty), \] or \[ X(t)=X'(-t) \text{ for all } t \in (-\infty,\infty) \text{ and } Y(t)=-Y'(t) \text{ for all } t \in [0,\infty). \] A similar result also holds when \(Z\) is replaced by \(X \circ X\). The second concern is an analogue of Khinchin’s law of the iterated logarithm. The author shows that a.s. \[ \limsup_{t \to 0+} {Z(t) \over t^{1/4} (\log | \log t |)^{3/4}}=2^{5/4}3^{-3/4}. \] (We mention that this last result can also be deduced from the law of the iterated logarithm for a stable subordinator with index 1/4 due to L. Breiman [Ann. Math. Stat. 39, 1818-1824 (1968; Zbl 0192.554)]).
For the entire collection see [Zbl 0780.00020].
Reviewer: J.Bertoin (Paris)

60J65 Brownian motion