Let U be an open set of ℝn, m be a positive
Radon measure on U such that , and
be a strongly continuous contraction sub-Markovian
semigroup on . We investigate the structure of
(i) Denote respectively by and the generator and the co-generator
of . Under the assumption that , we give an explicit Lévy–Khintchine type representation of A on .
(ii) If is an analytic semigroup and hence is associated with a semi-Dirichlet form , we give an explicit characterization of ℰ on under the assumption that .
We also present a LeJan type transformation rule for the diffusion part of
regular semi-Dirichlet forms on general state spaces.