Thick points for transient symmetric stable processes. (English) Zbl 0927.60077

Dembo, Amir; Rosen, Jay; Peres, Yuval; Zeitouni, Ofer

Electron. J. Probab. 4, Paper No. 10, 13 p. (1999).

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 \(\varepsilon \to 0+\). 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.

Reviewer: J.Bertoin (Paris)

MSC:

60J55

Local time and additive functionals

Keywords:

stable process; occupation measure; multifractal spectrum

Citations:

Zbl 0178.19301