Symmetric Markovian Semigroups and Dirichlet Forms

Abstract

This chapter studies the concepts of Dirichlet form and Dirichlet space by first working with a σ‎-finite measure space (E,B(E),m) without any topological assumption on E and establish the correspondence of the above-mentioned notions to the semigroups of symmetric Markovian linear operators. Later on the chapter assumes that E is a Hausdorff topological space and considers the semigroups and Dirichlet forms generated by symmetric Markovian transition kernels on E. The chapter also considers quasi-regular Dirichlet forms and the quasi-homeomorphism of Dirichlet spaces. From here, the chapter shows that there is a nice Markov process called an m-tight special Borel standard process associated with every quasi-regular Dirichlet form.