## Abstract

Let *U* be an open set of ℝ^{n}, *m* be a positive
Radon measure on *U* such that $\mathrm{supp}\left[m\right]=U$, and
${\left({P}_{t}\right)}_{t>0}$ be a strongly continuous contraction sub-Markovian
semigroup on ${L}^{2}(U;m)$. We investigate the structure of
${\left({P}_{t}\right)}_{t>0}$.

(i) Denote respectively by $(A,D(A\left)\right)$ and $(\widehat{A},D\left(\widehat{A}\right))$ the generator and the co-generator
of ${\left({P}_{t}\right)}_{t>0}$. Under the assumption that ${C}_{0}^{\infty}\left(U\right)\subset D\left(A\right)\cap D\left(\widehat{A}\right)$, we give an explicit Lévy–Khintchine type representation of *A* on ${C}_{0}^{\infty}\left(U\right)$.

(ii) If ${\left({P}_{t}\right)}_{t>0}$ is an analytic semigroup and hence is associated with a semi-Dirichlet form $(\mathcal{E},D(\mathcal{E}\left)\right)$, we give an explicit characterization of ℰ on ${C}_{0}^{\infty}\left(U\right)$ under the assumption that ${C}_{0}^{\infty}\left(U\right)\subset D\left(\mathcal{E}\right)$.

We also present a LeJan type transformation rule for the diffusion part of regular semi-Dirichlet forms on general state spaces.

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