We construct the Kuznetsov measure associated with an excessive measure, $m$, and the semigroup of a process killed according to a multiplicative functional, $M$, in terms of the Kuznetsov measure corresponding to $m$ and the original semigroup and certain functionals coming from $M$. We also obtain an explicit decomposition of $m$ into its invariant and purely excessive parts relative to the killed semigroup. This decomposition involves objects related to the fine structure of $M$ that have been introduced in recent years.
"Killing a Markov Process Under a Stationary Measure Involves Creation." Ann. Probab. 16 (2) 564 - 585, April, 1988. https://doi.org/10.1214/aop/1176991773