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 $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}[m]=U$, 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 $(\widehat{A},D(\widehat{A}))$ the generator and the co-generator of $(P_t{)}_{t>0}$. Under the assumption that $C_0^{\mathrm{\infty }}(U)\subset D(A)\cap D(\widehat{A})$, we give an explicit L'evy-Khintchine type representation of $A$ on $C_0^{\mathrm{\infty }}(U)$. (ii) If $(P_t{)}_{t>0}$ is an analytic semigroup and hence is associated with a semi-Dirichlet form $(\mathcal{E},D(\mathcal{E}))$, we give an explicit characterization of $\mathcal{E}$ on $C_0^{\mathrm{\infty }}(U)$ under the assumption that $C_0^{\mathrm{\infty }}(U)\subset D(\mathcal{E})$. We also present a LeJan type transformation rule for the diffusion part of regular semi-Dirichlet forms on general state spaces.