Abstract
We establish a uniform Hausdorff dimension result for the inverse image sets of real-valued strictly $\alpha $-stable Levy processes with $1< \alpha \le 2$. This extends a theorem of Kaufman [11] for Brownian motion. Our method is different from that of [11] and depends on covering principles for Markov processes.
Highlights
Let X = {X(t), t ≥ 0, Px} be a real-valued strictly α-stable Lévy process with α ∈
Stable Lévy processes form an important class of Markov processes
Without a local time, it is not clear to us how to construct a random Borel measure supported on X−1(F ) such that Frostman’s lemma is applicable
Summary
In the special case of F = {z}, it follows from Barlow et al [1, (8.7)] that if 1 < α ≤ 2 dimH X−1(z) = This gives a uniform Hausdorff dimension result for the level sets of X. To show the upper bound, we design a new covering principle (see Lemma 2.2 below) for the inverse images of recurrent processes ( it is applicable to α = 1). It follows from Hawkes and Pruitt [10] (see [22]) that the following uniform dimension result holds: Px (dimH X(E) = dimH E for all Borel E ⊂ R+) = 1.
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