Bivariate Revuz Measures and the Feynman-Kac Formula on Semi-Dirichlet Forms

In this paper we shall first establish the theory of bivariate Revuz correspondence of positive additive functionals under a semi-Dirichlet form which is associated with a right Markov process X satisfying the sector condition but without duality. We extend most of the classical results about the bivariate Revuz measures under the duality assumptions to the case of semi-Dirichlet forms. As the main results of this paper, we prove that for any exact multiplicative functional M of X, the subprocess X (M) of X killed by M also satisfies the sector condition and we then characterize the semi-Dirichlet form associated with X (M) by using the bivariate Revuz measure, which extends the classical Feynman-Kac formula.