The Transformation Of Dirichlet Forms And Its Associated Markov Processes

The Dirichlet form theory comes from classical potential theory in mathematical physics.According to the corresponding relationship between quasi-regular Dirichlet forms and Markov process,a bridge is built between classical potential theory and stochastic analysis.In this way,we can transform some analysis problems into stochastic analysis problems,so that Dirichlet form theory is widely used in the fields of potential theory,Markov processes,stochastic differential equations and many other related fields,and provides a strong theoretical basis for many mathematical problems.Therefore,it is of great practical significance to study the Dirichlet form theory.The transformation of Markov processed and its related Dirichlet forms have always been a research topic of great interest to mathematicians.From the point of view of Dirichlet forms,the study of the quasi-regularity of the new quadratic form obtained by Dirichlet forms transformation and its related Markov process can enrich the content of Dirichlet forms and Markov processes.The main work of this paper is as follows:For the Dirichlet forms of Brownian motion.Firstly,according to the relationship between the semigroup and the generator of Markov process,the generator expression of Brownian motion was obtained by Taylor expansion and other operations.Then,the Dirichlet forms corresponding to Brownian motion was obtained by using the relation between generator and quadratic form.Finally,we took the Dirichlet forms corresponding to Brownian motion as the basic form,and considered two kinds of transformations,one is to keep the reference measure unchanged and change the basic type,the other is to keep the basic form unchanged and change the reference measure.We also found the conditions that the Dirichlet forms transformation do not change its quasi-regularity.For the general symmetric quasi-regular Dirichlet forms.Firstly,we give the definition of Dirichlet forms transformation.Combined with the definition,we give the sufficient condition that the quadratic form after Dirichlet forms transformation is quasi-regular Dirichlet forms.Then,under this sufficient condition,through the one-to-one correspondence between symmetric quasi-regular Dirichlet forms and Markov processes,we discuss the Dirichlet forms transformation of the Dirichlet forms corresponding to the two order differential operator and the Pseudo differential operator respectively,and the Markov processes corresponding to the quasi-regular Dirichlet forms before and after transformation.Finally,we study the relationship between the two Markov processes through the characteristic function,and obtain that their characteristic functions differ by a logarithmic function,and this logarithmic function is related to the generators of quasi-regular Dirichlet forms before and after transformation.