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Spectral resolutions for correlation functions of some stochastic fields. (Russian. English summary) Zbl 0906.34041

The correlation function \(K(s,t)\) of 2-dimensional random fields \(x(t)\) satisfying a random linear operator differential equation is analyzed within the framework of a separable Hilbert space. The author presents a spectral resolution (representation) of the correlation function corresponding to that random field. The proof is based on certain properties of the Lie algebra of linear operators \(A_1, A_2\) with \([A_1,A_2] = i A_2\) and on the works of I. M. Gel’fand and M. A. Najmark [see, e.g., Mat. Sb., New Ser. 21(63), 405-434 (1947; Zbl 0038.01703)].

MSC:

34F05 Ordinary differential equations and systems with randomness
60H25 Random operators and equations (aspects of stochastic analysis)
47D03 Groups and semigroups of linear operators
93E03 Stochastic systems in control theory (general)

Citations:

Zbl 0038.01703
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