Perturbation theory for convolution semigroups

AbstractWe deal with convolution semigroups (not necessarily symmetric) in Lp(RN) 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 L1, we enlarge the extended Kato class potentials used in the theory of Markov processes. In L2 setting, Dirichlet form-perturbation theory is finely related to L1-theory and the extended Kato class measures is also enlarged. Finally, various perturbation problems for subordinate semigroups are considered