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Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space, disturbed by Gaussian processes. Part I

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Abstract

Nonlinear evolution differential equations with unbounded linear operators of disturbance by Gaussian random processes are considered in an abstract Hilbert space. For the Cauchy problem for the differential equations, the sufficient existence and uniqueness conditions for their solutions are proved and the sufficient conditions for the equivalence of the probability measures generated by these solutions are derived. Moreover, the corresponding Radon–Nikodym densities are calculated explicitly in terms of the coefficients or characteristics of the considered differential equations.

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Authors and Affiliations

  1. Donetsk Economic and Humanitarian Institute, Donetsk, Ukraine

    A. A. Fomin-Shatashvili

  2. Tugan-Baranovskii National University of Economics and Trade, Donetsk, Ukraine

    T. A. Fomina

  3. Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk, Ukraine

    A. D. Shatashvili

Authors
  1. A. A. Fomin-Shatashvili
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  2. T. A. Fomina
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  3. A. D. Shatashvili
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Correspondence to A. A. Fomin-Shatashvili.

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 89–101, November–December 2011.

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Fomin-Shatashvili, A.A., Fomina, T.A. & Shatashvili, A.D. Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space, disturbed by Gaussian processes. Part I. Cybern Syst Anal 47, 907–918 (2011). https://doi.org/10.1007/s10559-011-9370-y

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