On Regular Dirichlet Subspaces Of One-dimensional Diffusions

This thesis mainly discusses Dirichlet spaces of one-dimensional diffusions and their regular subspaces. We show that the ε₁-closure of an algebra of bounded functions in a Dirichlet space has Markovian property. This result gives a simple way to verify Markovian property. We also characterize all regular Dirichlet subspaces of Dirichlet space associated with a linear Brownian motion. The main results of this thesis include Theorem 1.1.1, Theorem 2.2.1, Theorem 2.3.1, Theorem 3.2.3 and Theorem 3.3.2. With the assumption that each point is regular, we characterize the local regular Dirichlet space of a symmetric diffusion on linear connected set I as follows: u is absolutely continuous with respect to swhere m is the reference measure, k is the killing measure, s is the scale function, C is a positive constant number. We construct the associated symmetric diffusion from a one dimensional (reflected or killed at the boundaries if finite) Brownian motion by a time change and a transform of state space if the killing measure k is zero. At the last, we give the sufficient and necessary conditions for a local regular Dirichlet form on L²(I, m) to be recurrent, transient and conservative.