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Thick Points for Transient Symmetric Stable Processes
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Amir Dembo, Stanford University
Yuval Peres, University of California, Berkeley
Jay Rosen, College of Staten Island, CUNY
Ofer Zeitouni, Technion
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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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Full text: PDF
Pages: 1-13
Published on: May 5, 1999
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Electronic Journal of Probability. ISSN: 1083-6489 |
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