Kato classes for L\'evy processes

TL;DR

This paper investigates the conditions under which different definitions of the Kato class for Lévy processes coincide, providing detailed characterizations and an analytic reformulation based on the process's characteristic exponent.

Contribution

It establishes the equivalence of Kato class definitions based on semigroup and resolvent for Lévy processes, and describes the classes when 0 is regular for 0, including an analytic reformulation.

Findings

The definitions coincide if and only if 0 is not regular for 0.

Detailed descriptions of the Kato classes when 0 is regular for 0.

An analytic reformulation using the Lévy-Khintchine exponent.

Abstract

We prove that the definitions of the Kato class by the semigroup and by the resolvent of the L\'{e}vy process on $\mathbb{R}^d$ 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 (L\'{e}vy-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.