| DING WANDING,CHEN MUFA.[J].数学年刊A辑,1981,2(1):47~59 |
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| QUASI-REVERSIBLE MEASURES OF NEARESTNEIGHBOUR SPEED FUNCTIONS |
| Received:December 10, 1979 |
| DOI： |
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| Let S be a countable set with a graph structure. The process with state space
\[X = {\{ 0,1\} ^s}\] is described in terms of a collection of nonnegative speed functions
\[c(u, \cdot ),u \in S\]. In this paper we introduce the concept of qnasi-reversible measure for
speed functions, and discuss some properties contained in the existence and uniqueness
of quasi-reversible measures for the nearest neighbour speed functions, with the idea of field theory by Hou and chen[3]； In section 2, we show that qnasi-reyerisible measures are Markov random fields. A necessary and sufficient condition for the
existence of quasi-reversible measures is presented. In seotion 3, a uniqueness theorem
of quasi-reversible measures is given. The problem to determine the quasi-reversible
measures in accordance with the speed function is discussed , for some particular cases,
the quasi-reversible measures can be computed explicitly. In seotion 4, we show that
if the speed functions are uniformly bounded, and each point of S has uniformly
bounded boundary then the quasi-reyersible measures of the speed functions are
reversible measures of the spin-flip process with the speed fnncfaons. Thus we obtain,
the necessary and sufficient conditions for the existence and uniqueness of reversible
measures for spin-flip process with nearest neighbour speed functions. Particularly if
speed functions are defined by the nearest neighbour potential, then quasi-reversible measure exigt，thus our results can be applied to solve the uniqueness problem of
Gibbs states with the nearest neighbour potential. |
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