Perturbation theory for convolution semigroups

Abstract

We deal with convolution semigroups (not necessarily symmetric) in L p ( R N ) and provide a general perturbation theory of their generators by indefinite singular potentials. Such semigroups arise in the theory of Lévy processes and cover many examples such as Gaussian semigroups, α-stable semigroups, relativistic Schrödinger semigroups, etc. We give new generation theorems and Feynman–Kac formulas. In particular, by using weak compactness methods in L 1 , we enlarge the extended Kato class potentials used in the theory of Markov processes. In L 2 setting, Dirichlet form-perturbation theory is finely related to L 1 -theory and the extended Kato class measures is also enlarged. Finally, various perturbation problems for subordinate semigroups are considered.