Extensions of Lévy–Khintchine formula and Beurling–Deny formula in semi-Dirichlet forms setting

The Lévy–Khintchine formula or, more generally, Courrège's theorem characterizes the infinitesimal generator of a Lévy process or a Feller process on Rd. For more general Markov processes, the formula that comes closest to such a characterization is the Beurling–Deny formula for symmetric Dirichlet forms. In this paper, we extend these celebrated structure results to include a general right process on a metrizable Lusin space, which is supposed to be associated with a semi-Dirichlet form. We start with decomposing a regular semi-Dirichlet form into the diffusion, jumping and killing parts. Then, we develop a local compactification and an integral representation for quasi-regular semi-Dirichlet forms. Finally, we extend the formulae of Lévy–Khintchine and Beurling–Deny in semi-Dirichlet forms setting through introducing a quasi-compatible metric.