Recurrence Criteria for Skew Products of Symmetric Markov Processes
Recurrence Criteria for Skew Products of Symmetric Markov Processes
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对称马尔可夫过程偏斜积的递归准则
DOI：
10.1515/form.1989.1.331
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发表时间：
1989
期刊：
影响因子：
--
通讯作者：
H. Ôkura
中科院分区：
文献类型：
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作者：
H. Ôkura
令M = {A？} （z = 1,2）是两个独立的循环对称马尔可夫过程，设A (t)是M的正连续加性泛函。我们将给出一个关于斜积过程M = {(A？*,Α^(ί})}。该检验是用对称马尔可夫过程的“速率函数”来表示的，类似于Tomisaki的检验[20]。这个证明是基于M. Fukushima和Y. Oshima[7]最近关于斜积过程的狄利克雷形式的表达式的结果的一个推广版本。1980数学学科分类（1985修订）：60J, 31C25。设{Xt} O ' = l»2)分别是局部紧可分度量空间X （i = l, 2）上的独立保守马尔可夫过程，设At是{X}*}的一个正连续加性泛函。那么过程Xt = (X}\ X£}称为{Α^}和{X$}对At的偏积。最近，福岛和大岛[7]已经确定了光滑流形上对称扩散的偏积的狄利克雷形式。他们还发现，如果At的Revuz测度或{X}}的速度测度有界，则循环扩散的偏积再次循环。本文通过引入递归马尔可夫过程的速率函数的概念，给出对称马尔可夫过程偏积递归的更一般的定量判据。在第1节中，我们将给出一些关于对称马尔可夫过程和狄利克雷形式理论的初步事实。偏积过程的递归准则将在第2节中建立，然后将在第3节中使用它来生成递归马尔可夫过程的几个具体示例。我们将更准确地解释内容。设M是X上的M对称马尔可夫过程，设（S, &）是它在L（x9m）上的狄利克雷形式。M定义为
Let M = {A?} (z = 1,2) be two independent recurrent Symmetrie Markov processes and let A (t) be a positive continuous additive functional of M. We will give an integral test for the recurrence of the skew product process M = {(A?*, Α^(ί})}. The test is formulated in terms of "rate functions" of Symmetrie Markov processes and analogous to Tomisaki's test [20]. The proof is based on a generalized Version of a recent result by M. Fukushima and Y. Oshima [7] concerning the expression of the Dirichlet form of the skew product process. 1980 Mathematics Subject Classification (1985 Revision): 60J, 31C25. Introduction Let {Xt} O ' = l » 2 ) be independent conservative Markov processes on locally compact separable metric spaces X (i = l, 2), respectively, and let At be a positive continuous additive functional of {X}*}. Then the process Xt = (X}\ X£} is called the skew product of {Α^} and {X$} with respect to At. Recently, Fukushima and Oshima [7] have determined the Dirichlet forms for skew products of Symmetrie diffusions on smooth manifolds. They have also found that the skew product of recurrent diffusions is recurrent again if either the Revuz measure of At or the speed measure of {X}} is bounded. In the present paper we will give more general quantitative criteria for the recurrence of skew products of Symmetrie Markov processes by introducing the notion of rate functions for recurrent Markov processes. In Section l we will give some preliminary facts on the theory of Symmetrie Markov processes and Dirichlet forms. Recurrence criteria for skew product processes will be established in Section 2, which will then be used in Section 3 to produce several concrete examples of recurrent Markov processes. We will explain the contents more precisely. Let M be an m-symmetric Markov process on X and let (S, &) be its Dirichlet form on L(X9 m). M is defined to be