Levy-Khintchine type representation of Dirichlet generators and Semi-Dirichlet forms

Let $U$ be an open set of $\mathbb{R}^n$, $m$ a positive Radon measure on $U$ such that ${\rm supp}[m]=U$, and $(P_t)_{t>0}$ a strongly continuous contraction sub-Markovian semigroup on $L^2(U;m)$. We investigate the structure of $(P_t)_{t>0}$. (i) Denote respectively by $(A,D(A))$ and $(\hat A,D(\hat A))$ the generator and the co-generator of $(P_t)_{t>0}$. Under the assumption that $C^{\infty}_0(U)\subset D(A)\cap D(\hat A)$, we give an explicit L\'evy-Khintchine type representation of $A$ on $C^{\infty}_0(U)$. (ii) If $(P_t)_{t>0}$ is an analytic semigroup and hence is associated with a semi-Dirichlet form $({\cal E}, D({\cal E}))$, we give an explicit characterization of ${\cal E}$ on $C^{\infty}_0(U)$ under the assumption that $C^{\infty}_0(U)\subset D({\cal E})$. We also present a LeJan type transformation rule for the diffusion part of regular semi-Dirichlet forms on general state spaces.