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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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##### References:
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