Abstract
We prove that the definitions of the Kato class through the semigroup and through the resolvent of the Lévy process in \(\mathbb {R}^{d}\) coincide if and only if 0 is not regular for {0}. If 0 is regular for {0} then we describe both classes in detail. We also give an analytic reformulation of these results by means of the characteristic (Lévy-Khintchine) exponent of the process. The result applies to the time-dependent (non-autonomous) Kato class. As one of the consequences we obtain a simultaneous time-space smallness condition equivalent to the Kato class condition given by the semigroup.
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Grzywny, T., Szczypkowski, K. Kato Classes for Lévy Processes. Potential Anal 47, 245–276 (2017). https://doi.org/10.1007/s11118-017-9614-1
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DOI: https://doi.org/10.1007/s11118-017-9614-1
Keywords
- Kato class
- Lévy process
- Lévy-Khintchine exponent
- Schrödinger perturbation
- Unimodal isotropic Lévy process
- Subordinator
- Polarity of a one point set