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1999 Thick Points for Transient Symmetric Stable Processes
Amir Dembo, Yuval Peres, Jay Rosen, Ofer Zeitouni
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Electron. J. Probab. 4: 1-13 (1999). DOI: 10.1214/EJP.v4-47


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 $c(b,d) \lt \sup_x \liminf_{r \to 0} T(x,r)/r^b \lt C(b,d)$.


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Amir Dembo. Yuval Peres. Jay Rosen. Ofer Zeitouni. "Thick Points for Transient Symmetric Stable Processes." Electron. J. Probab. 4 1 - 13, 1999.


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

zbMATH: 0927.60077
MathSciNet: MR1690314
Digital Object Identifier: 10.1214/EJP.v4-47

Primary: 60J55 , 60J55

Keywords: multifractal spectrum , occupation measure , Stable process


Vol.4 • 1999
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