Kato classes for Lévy processes. (English) Zbl 1379.60053
Authors’ 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.
Reviewer: Kun Soo Chang (Seoul)
MSC:
| 60G51 | Processes with independent increments; Lévy processes |
| 60J45 | Probabilistic potential theory |
| 47A55 | Perturbation theory of linear operators |
| 35J10 | Schrödinger operator, Schrödinger equation |
Keywords:
Kato class; Lévy process; Lévy-Khintchine exponent; Schrödinger perturbation; unimodal isotropic Lévy process; subordinator; polarity of a one point setReferences:
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