Geodesic
Markov Processes on Order-Unit Spaces
Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 1, pp. 153-162
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
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 
@article{TVP_2008_53_1_a8,
     author = {M. A. Berdikulov},
     title = {Markov {Processes} on {Order-Unit} {Spaces}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {153--162},
     year = {2008},
     volume = {53},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2008_53_1_a8/}
}
TY  - JOUR
AU  - M. A. Berdikulov
TI  - Markov Processes on Order-Unit Spaces
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2008
SP  - 153
EP  - 162
VL  - 53
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TVP_2008_53_1_a8/
LA  - ru
ID  - TVP_2008_53_1_a8
ER  - 
%0 Journal Article
%A M. A. Berdikulov
%T Markov Processes on Order-Unit Spaces
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2008
%P 153-162
%V 53
%N 1
%U http://geodesic.mathdoc.fr/item/TVP_2008_53_1_a8/
%G ru
%F TVP_2008_53_1_a8
M. A. Berdikulov. Markov Processes on Order-Unit Spaces. Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 1, pp. 153-162. http://geodesic.mathdoc.fr/item/TVP_2008_53_1_a8/

[1] Sarymsakov T. A., Polupolya i teoriya veroyatnostei, FAN, Tashkent, 1981, 89 pp. | MR

[2] Sarymsakov T. A., “Nekommutativnye veroyatnostnye prostranstva na $O^*$-algebrakh”, Dokl. AN SSSR, 241:2 (1978), 297–300 | MR | Zbl

[3] Sarymsakov T. A., Ayupov Sh. A., “Regulyarnost tsepei Markova na $O^*$-algebrakh”, Dokl. AN UzSSR, 1979, no. 4, 3–5 | MR | Zbl

[4] Ayupov Sh. A., “Ergodicheskie teoremy dlya tsepei Markova na $O^*$-algebrakh”, Dokl. AN UzSSR, 1978, no. 7, 11–13 | MR | Zbl

[5] Ayupov Sh. A., “Nezavisimost i markovskie protsessy v veroyatnostnykh prostranstvakh na iordanovykh algebrakh”, Predelnye teoremy dlya sluchainykh protsessov i smezhnye voprosy, FAN, Tashkent, 1982, 28–41 | MR

[6] Sarymsakov T. A., Zimakov N. P., “Ergodicheskii printsip dlya markovskikh polugrupp v uporyadochennykh normirovannykh prostranstvakh s bazoi”, Dokl. AN SSSR, 289:3 (1986), 554–558 | MR

[7] Alfsen E. M., Shultz F. W., Noncommutative spectral theory for affine functions spaces on convex sets, Mem. Amer. Mat. Soc., 172, AMS, Providence, R.I., 1976, 122 pp. | MR

[8] Berdikulov M. A., Odilov S., “Obobschennye spin-faktory”, Uzb. matem. zhurn., 1995, no. 1, 12–17 | MR

[9] Berdikulov M. A., Zhuraev I. M., “Normalnye polozhitelnye funktsionaly na prostranstvakh s poryadkovoi edinitsei”, Uzb. matem. zhurn., 1996, no. 4, 22–28 | MR | Zbl

[10] Sarymsakov T. A., Vvedenie v kvantovuyu teoriyu veroyatnostei, FAN, Tashkent, 1985, 184 pp. | MR

[11] Kholevo A. S., Veroyatnostnye i statisticheskie aspekty kvantovoi teorii, Nauka, M., 1980, 320 pp. | MR | Zbl

[12] Berdikulov M. A., “Novaya poryadkovaya struktura deistvitelnykh simmetrichnykh $(2\times2)$-matrits”, Vladikav. matem. zhurn., 6:4 (2004), 12–24 | MR