| Chen Mufa.[J].数学年刊A辑,1980,1(3-4):437~452 |
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| REVERSIBLE MARKOV PROCESSES IN ABSTRACT SPACE |
| Received:October 20, 1979 |
| DOI： |
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| Let \[(E,{\cal E})\] be a measurable space and every single point set {x} belong to
\[(E,{\cal E})\].\[q(x) - q(x,A)(x \in E,A \in {\cal E})\]is said to be a q-pair, if
(i) For fixed i，q(°), Л)\[A,q(),q(,A)\] is a \[{\cal E}\]-measurable function of x;
(ii) For fixed \[x,q(x, \cdot )\] is a measure on \[{\cal E}\], and
\[\begin{array}{l}
0 \le q(x,A) \le q(x,E) \le q(x) < \infty .(\forall x \in E,\forall A \in {\cal E})\q(x,\{ x\} ) = 0,(\forall x \in E)
\end{array}\]
A q-pair of furiotions q (x)- q (x,A) is called conservative when
\[q(x,E) = q(x),(\forall x \in E)\].
\[{P_t}(x,A)(t \ge 0,x \in E,A \in {\cal E})\] is said to be a q-process, if
(i) For fixed t, A, \[{P_t}(x,A)\] is a \[{\cal E}\]-measnrable function of x;
(ii)For fixed t,x, \[{P_t}(x, \cdot )\] is a measure on ê\[{\cal E}\] and \[0 \le {P_t}(x,E) \le 1\];
(iii) \[{P_{s + t}}(x,A) = \int_E {{P_t}} (x,dy){P_s}(y,A),{\kern 1pt} {\kern 1pt} {\kern 1pt} (x \in E,A \in {\cal E},t,s \ge 0)\]
(iv)\[\mathop {\lim }\limits_{t \to {0^ + }} \frac{{{P_t}(x,A) - {I_A}(x)}}{t} = q(x,A) - q(x){I_A}(x)(\forall x \in E,\forall A \in {\cal E})\]
It is called honest when
\[{P_t}(x,E) = 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} (\forall t \ge 0,\forall x \in E)\]
A q-process \[{{P_t}(x,A)}\] is called reversible, if there is a probability measure \[\mu \] on \[{\cal E}\] such,
that
\[\int_A {\mu (dx){P_t}} (x,B) = \int_B {\mu (dx){P_t}} (x,A){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (\forall t \ge 0.\forall A,B \in {\cal E})\]
In this paper, we obtain some oriterions for
(i)The existence of a reversible q-process;.
(ii)The existence of a honest reversible q-process；
(iii)The uniqueness of reversible q-process when the q-pair is conservative. |
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