Electronic Journal of Probability

Thick Points for Transient Symmetric Stable Processes

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

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Let $T(x,r)$ denote the total occupation measure of the ball of radius $r$ centered at $x$ for a transient symmetric stable processes of index $b \lt d$ in $R^d$ and $K(b,d)$ denote the norm of the convolution with its 0-potential density, considered as an operator on $L^2(B(0,1),dx)$. We prove that as $r$ approaches 0, almost surely $\sup_{|x| \leq 1} T(x,r)/(r^b|\log r|) \to b K(b,d)$. Furthermore, for any $a \in (0,b/K(b,d))$, the Hausdorff dimension of the set of ``thick points'' $x$ for which $\limsup_{r \to 0} T(x,r)/(r^b |\log r|)=a$, is almost surely $b-a/K(b,d)$; this is the correct scaling to obtain a nondegenerate ``multifractal spectrum'' for transient stable occupation measure. The liminf scaling of $T(x,r)$ is quite different: we exhibit positive, finite, non-random $c(b,d), C(b,d)$, such that almost surely $c(b,d) \lt \sup_x \liminf_{r \to 0} T(x,r)/r^b \lt C(b,d)$.

Article information

Electron. J. Probab., Volume 4 (1999), paper no. 10, 13 pp.

Accepted: 5 May 1999
First available in Project Euclid: 4 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J55: Local time and additive functionals 60J55: Local time and additive functionals

Stable process occupation measure multifractal spectrum

This work is licensed under aCreative Commons Attribution 3.0 License.


Dembo, Amir; Peres, Yuval; Rosen, Jay; Zeitouni, Ofer. Thick Points for Transient Symmetric Stable Processes. Electron. J. Probab. 4 (1999), paper no. 10, 13 pp. doi:10.1214/EJP.v4-47. https://projecteuclid.org/euclid.ejp/1457125519

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