Thick Points for Transient Symmetric Stable Processes
Yuval Peres (University of California, Berkeley)
Jay Rosen (College of Staten Island, CUNY)
Ofer Zeitouni (Technion)
Abstract
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 $b<d$ 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 $c(b,d)<\sup_x \liminf_{r \to 0} T(x,r)/r^b<C(b,d)$.
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Pages: 1-13
Publication Date: May 5, 1999
DOI: 10.1214/EJP.v4-47
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