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Keywords:
commutative nonstationary stochastic fields; correlation function; infinitesimal correlation function; contractive semigroup
commutative nonstationary stochastic fields; correlation function; infinitesimal correlation function; contractive semigroup
Summary:
The present paper is devoted to further development of commutative nonstationary field themes; the first studies in this area were performed by K. Kirchev and V. Zolotarev [4, 5]. In this paper a more complicated variant of commutative field with nonstationary rank 2, carrying into more general situation for correlation function is studied. A condition of consistency (see (7) below) for commutative field is placed in the basis of the method proposed in [4, 5] and developed in this paper. The following semigroup structures of correlation theory for disturbances and semigroups are used in this case: $T_t (\varepsilon )=\exp (it A_{\varepsilon })$, $A_\varepsilon = A_1 +\varepsilon A_2$, $|\varepsilon | \ll 1$.
References:
[1] Erdelyi, A. (ed.): Higher transcendental functions. McGraw-Hill, New York 1953. Zbl 0542.33002
[2] Kirchev, K. P.: On a certain class of non-stationary random processes. Teor. Funkts., Funkts. Anal. Prilozh., Kharkov 14 (1971), 150–160 (Russian).
[3] Kirchev, K. P.: Linear representable random processes. God. Sofij. Univ., Mat. Fak. 66 (1974), 287–306 (Russian).
[4] Kirchev, K. P., Zolotarev, V. A.: Nonstationary curves in Hilbert spaces and their correlation functions I. Integral Equations Operator Theory 19 (1994), 270–289. MR 1280124
[5] Kirchev, K. P., Zolotarev, V. A.: Nonstationary curves in Hilbert spaces and their correlation functions II. Integral Equations Operator Theory 19 (1994), 447–457. MR 1285492
[6] Livshits, M. S., Yantsevich, A. A.: Theory of operator colligation in Hilbert space. Engl. transl. J. Wiley, N.Y. 1979. MR 0634097
[7] Zolotarev, V. A.: On open systems and characteristic functions of commuting operator systems. VINITI 857-79, 1-37 (Russian).
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