Proof of Continuity and Symmetry of Feynman-Kac Kernels

Abstract

In this chapter we shall give a self-contained proof of Theorem 2.5. Readers, who are not interested in this proof at the moment, may continue with Chapter 4 where resolvent differences are considered. Because of its length, the proof of Theorem 2.5. got a separate chapter. The proof here is more general than the one in [231] and also the representations of the kernels are described in much more detail. It is also noticed that if the local Kato-Feller property of the positive part of V is replaced with a global property, then the proof of Theorem 2.5. is much simpler. This is so because the property

$$ \begin{gathered} \mathop {\lim \sup }\limits_{t \downarrow 0 x \in E} \int_0^t {ds} \int_E {dy p_0 } (s,x,y) V_ + (y) = 0, which then replaces \hfill \\ \mathop {\lim \sup }\limits_{t \downarrow 0 x \in E} \int_0^t {ds} \int_K {dy p_0 } (s,x,y) V_ + (y) = 0, \hfill \\ \end{gathered} $$

for all compact subsets K of E, enables us to use globally convergent Dyson expansions (see Appendix B, B.15) from the start. If we only require a local Kato-Feller property, then this need not be the case. We feel that a proof of our general result is justified for at least two reasons. The first one being that the present result is more general, than otherwise would be obtained and a second reason being that some of the techniques in our proof are interesting for their own sake.