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Cybernetics and Systems Analysis

, Volume 47, Issue 6, pp 907–918 | Cite as

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

  • A. A. Fomin-Shatashvili
  • T. A. Fomina
  • A. D. Shatashvili
Article

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.

Keywords

evolution differential equations evolution family of bounded operators Radon–Nikodym density equivalence of probability measures generating operator Hilbert–Schmidt operator 

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Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • A. A. Fomin-Shatashvili
    • 1
  • T. A. Fomina
    • 2
  • A. D. Shatashvili
    • 3
  1. 1.Donetsk Economic and Humanitarian InstituteDonetskUkraine
  2. 2.Tugan-Baranovskii National University of Economics and TradeDonetskUkraine
  3. 3.Institute of Applied Mathematics and Mechanics, National Academy of Sciences of UkraineDonetskUkraine

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