Construction of Markov Processes Associated With Quasi-Regular Dirichlet Forms

Abstract

In this chapter, we consider the construction of tight strong Markov processes associated with standard quasi-regular Dirichlet forms by using the language of nonstandard analysis. In Fan [154], the construction of strong Markov pro- cesses associated with standard symmetric Dirichlet forms was considered. In this chapter, we consider nonsymmetric Dirichlet forms. We consider a standard Dirichlet form (F(·, ·),D(F)) whose state space is a regular Hausdorff space Y. We impose conditions directly on the Dirichlet form (F(·, ·), D(F)) to obtain associated tight strong Markov processes. The motivation comes from the desire to find the relationship between the family of Dirichlet forms and the family of strong Markov processes.