Girsanov Transformation Of One Kind Of Strong Local Dirichlet Process

Dirichlet forms originated from the classical potential theory in mathematics and physics. In early1990s, Ma Z.M and his partner established one to one rela-tionship between regular Dirichlet forms and right continuous Markov process. The theory of Dirichlet forms is a convenient and applicable mathematical instrument which provides a bridge between classical potential theory and stochastic analysis, by using of this theory,we can interchange some analytical problems and stochastic problems to each other. Thus Dirichlet forms have applications in many related areas of potential theory,such as markov processes, stochastic differential equations, algorithms, quantum mechanics, quantum field theory and provide a strong theoret-ical foundation for many mathematical and physical problems. Therefore the study of Dirichlet forms has important practical significance. The transformation of pro-cess has always been the research topic which both mathematicians and physicists are interested in. We can gain the new process and its associated Dirichlet form by the transformation of process. The discussion of the new process and its associated Dirichlet form, to a large extent,can enrich the contents of the Dirichlet forms and processes. Due to the close link between Girsanov transformation and the absolute continuity of the process, many scholars are interested in studying Girsanov trans-formation of process and have made many important research results. Let (ε, D(ε)) be a Dirichlet form, it is well known that for u∈D(ε), u(Xt) admits the following Fukushima’s decomposition:where Ntu is a CAF’s of zero energy. Chen Chuanzhong in [4,6] discussed the following type of generalized Feynman-Kac semigroup:The main difficulty in studying Feynman-Kac semigroup Ptu is the CAF’s of zero en-ergy Ntu may be of infinite variation. The author successfully translated the infinite variations case into a finite variations case by three methods:Girsanov transfor-mation of the processes Xt, perturbation of Dirichlet forms and h-transformation of quadratic forms,and also obtained a sufficient and necessary condition for Ptu to be strongly continuous. This article generalizes the results in [5] about Girsanov transformation of the processes Xt(from bounded function u to unbounded function u) and the killing measure of process remains exist under Girsanov transformation. However,properties (such as transient,recurrent and irreducible) of the new Dirichlet form associated with the transformed process are seldom discussed.In this dissertation, we mainly focus on the Girsanov transformation of stochas-tic process associated with one dimensional strong local Dirichlet form.We principal-ly discuss the properties of the transformed process and its associated Dirichlet form Firstly,we give the concrete expression of transformed Dirichlet Forms (ε, D(ε))for bounded u∈D(ε) and prove that I has no ε-polar set which consequently implies the quasi-continuity of a fuction is equivalent to its ordinary continuity.Also,concrete expression of transformed Dirichlet form for more general u is given. Secondly,when both r1, r2are regular,we obtain u is bounded,then (ε, D(ε)) is a regular,strongly local,recurrent and irreducible Dirichlet forms on L2([r1, r2];1I·m).When both r1,r2are approachable but non-regular,we obtain that u is bounded and (ε,D(ε)) is a regular,strongly local,transient and irreducible Dirichlet form on L2(I;m). Also,we consider the properties of corresponding new process. Let X(t) satisfies the stochas-tic differential equation Define the operator of X(t) as follow:In fact,the L here is the corresponding generator of the Dirichlet form associated with one dimensional diffusion process, of which we study the Girsanov transforma-tion in this article. The operator has a very important role in the study of stochastic differential equations. Many researchers are more interested in the operator or the coefficients of stochastic differential equations. By observing the changes in the expressions of Dirichlet forms associated with the processes before and after trans-formation, we give the expression of the generator associated with the transformed forms This result shows the application of Dirichlet form in stochastic diferential equations...