Uniform Dimension Results for the Inverse Images of Symmetric Lévy Processes

Abstract

We prove uniform Hausdorff and packing dimension results for the inverse images of a large class of real-valued symmetric Lévy processes. Our main result for the Hausdorff dimension extends that of Kaufman (1985) for Brownian motion and that of Song, Xiao, and Yang (2018) for α-stable Lévy processes with 1 < α < 2. Along the way we also prove an upper bound for the uniform modulus of continuity of the local times of these processes.

Hitting probabilities for Lévy processes on the real line

“…To our best knowledge, such results have not been obtained before. Apart from its own value, they can be used, together with heat kernel estimates (Grzywny and Szczypkowski, 2020) for instance for estimation of the Hausdorff dimension of the inverse images of Lévy processes (see Park et al, 2020). We also remark that although our main object to operate with is the real part of the characteristic exponent, one can work with the tail of the Lévy measure instead, since in view of Grzywny et al (2018, Proposition 3.8), scaling property of the latter implies scaling of the former.…”

Singular integrals of subordinators with applications to structural properties of SPDEs

“…b = 0 and ν(ds) = αΓ(1 − α) −1 s −α−1 ds. We refer the reader to [2,3,4,7,12,15,20] for results on α-stable subordinate Brownian motion.…”

Hitting probabilities for Lévy processes on the real line

“…To our best knowledge, such results have not been obtained before. Apart from its own value, they can be used, together with heat kernel estimates ( [13]) for instance for estimation of the Hausdorff dimension of the inverse images of Lévy processes (see [21]). We also remark that although our main object to operate with is the real part of the characteristic exponent, one can work with the tail of the Lévy measure instead, since in view of [10,Proposition 3.8], scaling property of the latter implies scaling of the former.…”