| Abstract: | 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 + sumlimits_{k in K} {{r_{ij}}^{(k)}} + sumlimits_{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 [sumlimits_{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 [sumlimits_{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. |
