Description of Markovian Random Fields by Gibbsian Conditional Probabilities

M. B. Averintsev

Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 1, pp. 21-35

Résumé

Let $T$ be a $v$-dimensional cubic lattice and $L$ a finite set of points from $T$. Suppose that the conditional probabilities of a random field $\xi(t)$ are positive and for any $s\in T$, $x$, $x(t)$. $\Pr\{\xi(s)=x\mid\xi(t)=x(t),\ t\in T\setminus\{s\}\}=\Pr\{\xi(s)=x\mid\xi(t)=x(t),\ t\in L+s\}$ Then $\xi(t)$ is called an $L$-Markov random field with positive conditional probabilities. In the paper, we prove that any such field $\xi(t)$ is a Gibbs field, in general, with many-particle potential.