Slow points and fast points of local times
type: Item
instance of: scholarly article
title: Slow points and fast points of local times
zbMATH Open document ID: 0945.60069
author: Laurence Marsalle
published in: The Annals of Probability
publication date: 1999-11-09T00:00:00Z
review text: 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 \(1\), \[ \forall t\in\mathbb{Z}^0\;\limsup_{s\to 0}|L_{t+ s}- L_t|\varphi(1/|s|)> 0,\quad \exists t\in \mathbb{Z}^0\;\limsup_{s\to 0}|L_{t+ s}- L_t|\varphi(1/|s|)< \infty. \]
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 1359643
zbMATH Keywords: modulus of continuity
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: slow points
zbMATH Keywords: local time
reviewed by: Ion C. Vladimirescu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677257
OpenAlex ID: W2083196318
cites work: Q3975586
cites work: Some applications of subordinators to local times of Markov processes
cites work: Q4726487
cites work: Q5556844
cites work: On the oscillation of the Brownian motion process
cites work: Uniform local behavior of stable subordinators
cites work: Lower functions for increasing random walks and subordinators
cites work: Uniform lower functions for subordinators
cites work: Uniform dimension results for processes with independent increments
cites work: Lower tail probability estimates for subordinators and nondecreasing random walks
cites work: Q5184252
cites work: Q4118621
cites work: Q3959913
cites work: How Often on a Brownian Path Does the Law of Iterated Logarithm Fail?
cites work: Logarithmic multifractal spectrum of stable occupation measure
cites work: The exact hausdorff measure of the zero set of a stable process
DOI: 10.1214/AOP/1022677257
description: scientific article; zbMATH DE number 1359643