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

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Let U be an open set of Rⁿ, m a positive Radon measure on U such that supp[m]=U, and (Pₜ)ₜ>₀ a strongly continuous contraction sub-Markovian semigroup on L²(U;m). We investigate the structure of (Pₜ)ₜ>₀. (i) Denote respectively by (A,D(A)) and (A,D(A)) the generator and the co-generator of (Pₜ)ₜ>₀. Under the assumption that C∞₀(U)⊂ D(A)∩ D(A), we give an explicit Lévy-Khintchine type representation of A on C∞₀(U). (ii) If (Pₜ)ₜ>₀ is an analytic semigroup and hence is associated with a semi-Dirichlet form (E, D(E)), we give an explicit characterization of E on C∞₀(U) under the assumption that C∞₀(U)⊂ D(E). We also present a LeJan type transformation rule for the diffusion part of regular semi-Dirichlet forms on general state spaces.