Perturbation Of Dirichlet Forms And Feynman-Kac Semigroups

The theory of Dirichlet forms is a convenient and applicable mathematical instrument which provides a bridge between classical potential theory and stochastic analysis, by the mean of this theory we can interchange some analytical problem and stochastic problem to each other. The theory of Feynman-Kac semigroups is an important subject closely related to Dirichlet forms and many other branch of mathematics and physics . Let be a Dirichlet form, it is well known that for admits the following Fukushima's decomposition:where N is a CAF's of zero energy . In this thesis we discuss the following type of generalized Feynman-Kac semigroup:In the next chapter, we study the strong continuity of the generalized Feynman-Kac semigroup P, obtain a sufficient and necessary condition for P to be strongly continuous . Our result improves partly the earlier works e.g. by Chen and Zhang [15] and by J. Glover et [31] in which the authors provided only sufficient conditions for the strong continuity of P . we study in detials the three methods to translate the infinite variations case into a finite variations case. The three metholds are: Girsanov transformation of the processes Xt, perturbation of Dirichlet forms and h-transformation of quadratic forms. We first generalize the results in [15] about Girsanov's transforms of the processes Xt( from bounded function u to unbounded function u), and describe the Dirichlet form associatedwith the transformed processes Xt At the end of this chapter, we give some examples of u such that is not in Kato class.In the third chapter, we discuss first the perturbation of a non-symmetric Dirichlet form with signed smooth measure , and get a sufficient condition for the quadratic form to be closable and semibounded. Then we study Girsanov's transformations of the pair of Markov processes as-sociated with . Given , denote by the transformed processes of . we discover that are not necessarily in duality with respect to the measure . This is different from the case of symmetric Dirichlet processes, and this raises more difficult and more interesting questions to be studied. We obtain a sufficient and necessary condition for to be in duality with respect to the measure . At last, we give two examples about the Girsanov transformation of non-symmetric Dirichlet processes, one of whicn tells us that are in duality with respect to the measure , and another says that are not in duality with respect to the measure .