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.Applied scientific and engineering problems are known to involve differential equations with random terms or coefficients that describe the behavior of systems in random media. The solutions of such equations obviously generate probability measures in infinite-dimensional spaces.One of the major problems in the theory of random processes is to establish the sufficient conditions for the equivalence of the above-mentioned probability measures with respect to some standard well-studied measures and to explicitly determine the corresponding Radon-Nikodym densities, naturally, in terms of known quantities, and in our case, in terms of the coefficients of the equations under study, their characteristics or abstract transformations. The existence of such densities allows an efficient solution of applied problems with the use of integration over known standard distributions (for example, Gaussian, Wiener, etc.) for which algorithms are available.The publications [1-42] address the absolute continuity and equivalence of probability measures for different classes of nonlinear transformations and nonlinear differential equations with Gaussian perturbation, which were applied to calculate optimal estimates in extrapolation and filtration problems for the solution of the differential equations [7,8,12,[34][35][36][37][38][39].The present paper can be considered as a continuation of the studies in this field for nonlinear evolution differential equations in a Hilbert space H and of the studies initiated in [44] for random fields being the solutions of Dirichlet and Neumann boundary-value problems for nonlinear elliptic differential equations in a finite-dimensional Euclidean space.In contrast to the previous studies, we will consider, for the first time, nonlinear differential equations with unbounded linear operators, which are a family of generating operators for the evolution family of bounded operators. We will use the concept of extended stochastic integral and the results from [9] to establish the sufficient conditions for the equivalence of the probability measures under study and to calculate explicitly the corresponding Radon-Nikodym densities.Denote by { } W, , Ã P a fixed probability space and by H a real separable Hilbert space with scalar product ( ,...

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In the first part of this paper it is received the closed general expression which can be considered as algorithm for obvious calculation of an optimum estimation in problems of a filtration of random processes accepting the values in abstract Hilbert spaces. Using a condition of existence of density of Radon-Nikodym's measure generated by random process concerning some Gaussian measure (the optimum filter) it is shown that calculation of conditional mathematical expectation is reduced to calculation of an unconditional population mean on the base of obvious expressions.In this paper, we continue the studies, initiated in [1, 2], with a focus on optimal filtering problems. Let us give a general setting of the problem considered here. Consider a random process in a Hilbert space H, simultaneously containing two processes, an unobservable process, which is the one we are interested in and is called a "useful signal", and the second process, called the "random noise", superposed with the "useful signal" distorting or completely contained in it. The optimal filtering problem consists in the following: using observable values of the process under consideration of any of its components, to separate the "random noise" and the "useful signal", and to construct, in some sense, the best approximation for the unobserved process, i.e., the "useful signal", possibly with time anticipation.The paper consists of two parts. In the first part, we will consider a general setting of the problem in an abstract Hilbert space H and obtain general formulas and algorithms for an explicit calculation of optimal estimates in filtration problems.In the second part, the obtained algorithms will be used for finding, in an explicit form, optimal estimates in the filtration problems for systems of nonlinear evolution differential equations perturbed with Gaussian random processes in a Hilbert space H. In the case where these equations contain a small nonlinearity, the obtained estimates are expanded in powers of the small parameter at the nonlinear term; the principal term Brought to you by | HEC Bibliotheque Maryriam ET J. Authenticated Download Date | 6/8/15 5:30 AM

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