INTEGRAL QUADRATIC FUNCTIONAL BASED ON A TWO-DIMENSIONAL MARKOV 2D FIELD AND ITS STATISTICS

This paper investigates the generating function of the distribution of an integral quadratic functional constructed on the basis of the Ornstein – UhlenbeckMarkov process. The object of the study is a two-dimensional Ornstein – Uhlenbeck process considered as a real Gaussian two-dimensionalMarkov field, together with the integral quadratic functional defined on this field. The subject of the research comprises the statistical characteristics ofthe functional, its distributional properties, and the peculiarities of its behavior depending on the parameters of the random field. In the analysis of such objects, the study of integral characteristics plays a crucial role, as they provide generalized information about the structure and properties of a random field over a given observation domain. One of the most informative characteristics is the integral quadratic functional, whose values reflect the energyand statistical properties of the underlying random process or field. The paper considers a twodimensional (2D) Markov field and the corresponding integral quadratic functional constructed on its basis. The generating function of the distribution of this functional is investigated, making it possible to obtain complete information about its probabilistic characteristics and moments of different orders. The analysis of the generating function reveals the distributional patterns of the functional values and identifies the distinctive features of its statistical behavior. For the case under consideration, an explicit analytical expression for the generating function of the distribution is derived, and a formula for the integral quadratic functional itself is obtained. Particular attention is devoted to extending the results to the case of stationary Gaussian multidimensional Markov fields. The proposed approach enables the methodology to be generalized to a broader class of random fields and processes possessing Markov properties. The obtained resultscan be applied to problems of statistical estimation, parameter identification of random fields, spatial data analysis, and the development of mathematical models for complex stochastic systems. The practical significance of the study lies in the possibility of applying the developed analytical methods to the analysis and processing of two-dimensional random signals and images, as well as to the further development of the theory of Markov random fields and their functionals.