Thick points for transient symmetric stable processes
type: Item
instance of: scholarly article
title: Thick points for transient symmetric stable processes
zbMATH Open document ID: 0927.60077
author: Amir Dembo
author: Yuval Peres
author: Jay Rosen
author: Ofer Zeitouni
published in: Electronic Journal of Probability
publication date: 1999-07-08T00:00:00Z
full work available at URL: https://eudml.org/doc/120048
full work available at URL: paper9.abs.html
review text: Precise information on the local rate of growth of a Brownian path are given by Khinchin's law of the iterated logarithm (growth rate at a fixed point), Lévy's modulus of continuity (maximal growth rate), the result by Orey and Taylor on the dimension of fast points (points at which the path growths faster than usual), and its counterpart for slow points. The purpose of the paper is to provide analogous results for the occupation measure of transient symmetric stable processes in \({\mathbb R}^d\); thereby extending previous results by the authors in the Brownian case. More precisely, let \(\left(X_t, t\geq 0\right)\) denote the stable process and \(\beta\in (0,2]\) its index, and introduce \[ \mu^X_T\left(B(x,\varepsilon)\right)=\int_{0}^{T}\mathbf{1}_{\{| X_t-x| \leq \varepsilon\}}dt , \] that is the total time spent by \(X\) until time \(T\in (0,\infty]\) in the ball centered at \(x\in \mathbb R^d\) with radius \(\varepsilon>0\). The quantity of interest is the asymptotic behavior of \(\mu^X_T\left(B(x,\varepsilon)\right)\) as \(\varepsilon \to 0+\). \textit{S. J. Taylor} [J. Math. Mech. 16, 1229-1246 (1967; Zbl 0178.19301)] has obtained the law of the iterated logarithm, viz. \[ \limsup_{\varepsilon\rightarrow 0+}{\mu^X_T\left(B(0,\varepsilon)\right)\over \varepsilon^{\beta}\log| \log \varepsilon| }=c\qquad \text{a.s.} \] for some positive and finite constant \(c\). The first result of the paper gives the analogue of Lévy's modulus of continuity: \[ \lim_{\varepsilon\rightarrow 0+}\sup_{x\in {\mathbb R}^d}{\mu^X_T\left(B(x,\varepsilon)\right)\over \varepsilon^{\beta} | \log \varepsilon| }=2c\qquad \text{a.s.} \] Then the Hausdorff dimension of the set of so-called thick points, that is points \(x\in {\mathbb R}^d\) such that \[ \lim_{\varepsilon\rightarrow 0+}{\mu^X_T\left(B(x,\varepsilon)\right)\over \varepsilon^{\beta} | \log \varepsilon| }=a , \] is computed, and some further results in this vein are given.
reviewed by: Jean Bertoin
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 1309966
zbMATH Keywords: stable process
zbMATH Keywords: occupation measure
zbMATH Keywords: multifractal spectrum
MaRDI profile type: Publication
OpenAlex ID: W2090935110
DOI: 10.1214/EJP.V4-47
description: scientific article; zbMATH DE number 1309966