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.