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Slow points and fast points of local times. (English) Zbl 0945.60069
Let \(L= (L_t, t\geq 0)\) be a local time, \(X_t= \inf\{u> 0: L_u> t\}\), \(t\geq 0\), and \(\varphi\) its Laplace exponent, which is specified by \(E[\exp- \lambda X_t]= \exp- t\varphi(\lambda)\), \(t\geq 0\), \(\lambda\geq 0\). The author proves the following theorems:
1. Suppose that \(\varphi\) is regularly varying at infinity with index \(\alpha\in (0,1)\) and that \(\beta> 0\). Then with probability \(1\), \[ \dim\Biggl\{ t\in (0,1): \limsup_{s\to 0} {(L_{t+ s}- L_t)\varphi(s^{-1}|\log s|)\over t\to 0^+}\geq \beta c_\alpha\Biggr\}= \alpha(1- \beta^{1/(1- \alpha)}), \] where dim denotes the Hausdorff dimension and with the convention that \(\dim E< 0\) if and only if \(E= \emptyset\).
2. If there exists some \(\lambda> 1\) such that \(1< \liminf_{x\to+ \infty} {\varphi(\lambda x)\over \varphi(x)}\leq \limsup_{x\to+ \infty} {\varphi(\lambda x)\over \varphi(x)}< \lambda\), one has that with probability \(1\), \[ \forall t\in\mathbb{Z}^0\;\limsup_{s\to 0}|L_{t+ s}- L_t|\varphi(1/|s|)> 0,\quad \exists t\in \mathbb{Z}^0\;\limsup_{s\to 0}|L_{t+ s}- L_t|\varphi(1/|s|)< \infty. \]

MSC:
60J55 Local time and additive functionals
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