Perturbations of Free Feller Operators

Abstract

There are several ways of introducing semigroups with perturbed generators. The analytic way starts with the unperturbed semigroup and uses the Trotter-product formula to find a Feynman-Kac representation of the perturbed semigroup. The semi-analytic or semi-stochastic manner begins again with the unperturbed semi-group. Then the potentials are introduced stochastically by verifying the sensibility and the semigroup property of the Feynman-Kac formula. Here we employ a purely stochastic approach in the sense that we begin with the transition density function of a strong Markov process. Our aim is to formulate all assumptions on the process or its generator in terms of assumptions on the density. In fact these kernels are the one-dimensional transition densities (marginals) of a strong Markov process. We assume that these transition densities possess a number of basic properties given in BASSA. An advantage, as described in the previous chapter, is that we are able to consider a large class of generators of Feller semigroups. Among others this class contains all the examples of section C in Chapter 1. The regular and singular perturbations are introduced stochastically via the Feynman-Kac formulae. In these formulae we use in an essential way the (strong) Markov processes generated by the unperturbed (free) Feller operator. Other stochastic methods (like martingale theory) will also allow us to describe the integral kernels for the corresponding Feynman-Kac semigroups. This can be done for regular as well as singular perturbations. The final objective is to present and construct a spectral theory for these free and perturbed Feller operators. For the reasons mentioned above this theory is called “Stochastic Spectral Theory”.