**Abstract:** | I will talk on the coincidence of two classes of Green-tight smooth measures of Kato class for general symmetric Markov processes. This coincidence has been known for (resolvent) strong Feller processes ((RSF) in short) by Kim and myself. We extend it for general symmetric Markov process satisfying the absolute continuity condition for transition probability ((AC) in short). We have an example satisfying (AC) but not (RSF).As a consequence of this coincidence, we can establish the compact embedding of the domain of Dirichlet forms and the existence of ground state provided the underlying measure is a Green-tight smooth measure of Kato class for $1$-subprocess and the given process satisfies (AC), which extends the recent seminal work on the compact embedding theorem by Takeda. This is a joint work with Kaneharu Tsuchida (National Defense Academy). If I have a time, I can mention the large deviation principle for the triple $(A_t^{?mu},A_t^F,N_t^u)$ of additive functionals as an application of local compact embedding theorem in this generality, which will be a joint work with Zhen-Qing Chen and Kaneharu Tsuchida. |