Markov Processes on Order-Unit Spaces

M. A. Berdikulov

Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 1, pp. 153-162

Résumé

The present paper studies Markov operators on order-unit spaces. The examples of these spaces in the commutative case are $M$-spaces, in the noncommutative case are the Hermitian part of $C*$- or $W*$-algebras, and in the nonassociated case are JB- or JBW-algebras. The Markov operators on these objects were studied by Sarymsakov [Semifields and Probability Theory, FAN, Tashkent, 1981 (in Russian)], [Dokl. Akad. Nauk SSSR, 241 (1978), pp. 297–300 (in Russian)]; Sarymsakov and Ajupov [Dokl. Akad. Nauk UzSSR, 1979, pp. 3–5 (in Russian)]; and Ajupov [Dokl. Akad. Nauk UzSSR, 7 (1978), pp. 11–13 (in Russian)], [Limit Theorems for Random Processes and Related Problems, FAN, Tashkent, 1982, pp. 28–41 (in Russian)]. On duality spaces for order-unit spaces, i.e., on spaces with basis norm, Markov operators were studied by Sarymsakov and Zimakov [Dokl. Akad. Nauk SSSR, 289 (1986), pp. 554–558 (in Russian)]. The given paper introduces a regular Markov process and proves the regularity properties for processes satisfying the condition analogous to condition (A) considered by Sarymsakov [Dokl. Akad. Nauk SSSR, 241 (1978), pp. 297–300 (in Russian)]. We prove the theorems connected with the regularity, accuracy, and periodicity of the Markov operator, study the Markov operators on matrix spaces which are not algebras, and find the general form of the Markov operator and sufficient conditions for the Markov property of linear operators.

Détail

Keywords: order-unit space, Markov operator, Markov process, regularity, periodicity, state, space of closed Hermitian $(2\times 2)$-matrices, $p$-order, $p$-eigenvalues.
Mots-clés : Markov chain