Some theorems on time change and killing of Markov processes

It is known that a Markov process is transformed to another Markov process by its continuous non-negative additive functional φt through time change or killing. υ On the other hand, φt is determined by an excessive function u(a)=Ma[φζ-o]. Moreover, if the Green measure of the process is expressed by g(a, b)m(db) and if the process satisfies some additional conditions, then u has the Riesz representation: u(ά)=$g(a, b)n(dV) with some measure n. These results are found in the works of Hunt [4], Volkonskii [13] and Meyer [9] under a general setup and in McKeanTanaka [7] in a concrete case. We want to study what meaning the measure n or m has for the process obtained through time change or killing. In the course of the study we need various generalizations of the resolvent equation and we are compelled to give a unified form in their treatments which is given in §2. In §3 we state construction theorems of processes by time change and killing and give some lemmas concerning (sub) invariant measures. Further, it is proved that the terminal measure of the killed process is represented by K\ (defined in § 2) and that a measure n is the terminal measure of the killed process with initial measure n if and only if n is an invariant measure of the process obtained through time change. In §4, Gλa and K λ a, defined in §2, are represented using a kernel function gϊ(a, b) under some regularity conditions for the Green kernel ga(a, b). In § 5 we prove that the Riesz measure n is a subinvariant measure of the process obtained through time change by the corresponding additive functional and give some sufficient conditions for the measure to be invariant. And also the meanings of n for killed process are discussed. In order to obtain a necessary and sufficient condition for the measure n to be invariant, we need some considerations on the adjoint process of the process obtained through time change or killing, which is given in § 6. The necessary and sufficient condition is stated in §7. The adjoints of the processes are also treated in [4], [8], [11] and [13].