Some theorems on time change and killing of Markov processes
Some theorems on time change and killing of Markov processes
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关于时间变化和马尔可夫过程消杀的一些定理
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发表时间：
1963
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通讯作者：
Ken
中科院分区：
文献类型：
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作者：
M. Nagasawa;Ken
已知一个马尔可夫过程是由其连续的非负加性泛函φt经过时间变化或杀戮而转化为另一个马尔可夫过程。另一方面，φt由一个过量函数u(a)=Ma[φζ-o]决定。此外，如果过程的Green测度用g(a, b)m（db）表示，并且如果过程满足一些附加条件，则u具有Riesz表示：u（ά）=$g(a, b)n（dV）具有一些测度n。这些结果在Hunt [4]， Volkonskii[13]和Meyer[9]在一般情况下和McKeanTanaka[7]在具体情况下的著作中都有发现。我们想研究n或m对于通过时间变化或杀戮得到的过程有什么意义。在研究过程中，我们需要对分解方程作各种各样的推广，而我们不得不在它们的处理中给出一个统一的形式，如§2所示。在§3中，我们给出了时间变化和杀戮过程的构造定理，并给出了一些关于（次）不变测度的引理。进一步证明了终止过程的终端测度用§2定义的K表示，且当且仅当n是通过时间变化得到的过程的不变测度时，一个测度n就是初始测度为n的终止过程的终端测度。在§4中，在Green核ga（a, b）的正则性条件下，用核函数gκ (a, b)表示了§2中定义的Gλa和K λa。在§5中，我们证明了Riesz测度n是由相应的加性泛函通过时间变化得到的过程的次不变测度，并给出了该测度是不变的一些充分条件。并讨论了n对于终止进程的意义。为了得到测度n不变的充分必要条件，我们需要考虑在§6中给出的通过时间变化或杀戮得到的过程的伴随过程。充分必要条件见§7。这些过程的伴随也在[4]、[8]、[11]和[13]中进行处理。
It is known that a Markov process is transformed to another Markov process by its continuous non-negative additive functional φt through time change or killing. υ On the other hand, φt is determined by an excessive function u(a)=Ma[φζ-o]. Moreover, if the Green measure of the process is expressed by g(a, b)m(db) and if the process satisfies some additional conditions, then u has the Riesz representation: u(ά)=$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].