| HOU ZHENTING,WANG PEIHUANG.[J].数学年刊A辑,1980,1(1):139~148 |
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| THE DECOMPOSITION THEOREM OF A PROBABILITY-FLOW |
| Received:November 06, 1978 |
| DOI： |
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| 英文摘要: |
| suppose that p is a Markov transition matrix on the sapce E,and {ui}(\[i \in E\])is an initial distribution.The Matrix (ui,pij)is called a probility-flow.we obtain the following theorem:For any initial distribution {ui}(ui>0)which need not be stationary,we have
\[{u_i}{p_{ij}} = {u_i}{p_{ij}}^d + \sum\limits_{k \in K} {{r_{ij}}^{(k)}} + \sum\limits_{i \in L} {{g_{ij}}^{(l)}} \]
where,
1) \[{u_i}{p_{ij}}^d = {u_i}{p_{ij}}^d(i,j \in E)\]
\[{p_{ij}}^d\]is called the detailed balabce part of p;
2)For each \[k \in K\](at most denumerable),there is a circular road
\[{a^{(k)}} = (i_1^{(k)},i_2^{(k)},...,i_n^{(k)},i_1^{(k)})\](\[n \geqslant 3,{i_s} \ne {i_t}(S \ne t,1 \leqslant S,t \leqslant n\]),and there is a constant \[{c_k} > 0\],such that
\[{r_{ij}}^{(k)} = \left\{ {\begin{array}{*{20}{c}}
{{c_k},(i,j) \in {a^{(k)}}} \\
{0,(else)}
\end{array}} \right.\]
and \[\sum\limits_{k \in K} {{r_{ij}}^{(k)}} \] is called the circulation part of p;
3)For any \[l \in L\](at most denumerable),there is a read in E;
\[{r^{(l)}} = (j_1^{(1)},...,j_n^{(l)})\]
\[n \geqslant 2,{j_s}^{(l)} \ne {j_t}^{(l)}(s \ne t,l \leqslant s,t \leqslant n)\],and there is a constant \[{d_l} > 0\],such that
\[{g_{ij}}^{(l)} = \left\{ {\begin{array}{*{20}{c}}
{{d_l},(i,j) \in {r^l}} \\
{0,(else)}
\end{array}} \right.\]
and \[\sum\limits_{i \in L} {{g_{ij}}^{(l)}} \]is called the divergent part of p.
This theorem is extetion of the theorem of circulation decomposition given by Qian Minping. |
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