Abstract. Generalized diffusion processes can serve as mathematical models of physical phenomenon of diffusion in media with the membranes on some fixed surfaces. Qualitatively, such processes differ from the ordinary diffusion process only at the points of location of the membrane, whose type is specified by the corresponding variants of the general Feller-Wentzell boundary condition (one-dimensional case) and the Wentzell condition (multidimensional case). An important problem in the theory of diffusion processes is the development of methods of construction of the process by the given diffusion characteristics: the diffusion matrix and the drift vector, and in the case of the state space with the membrane, by the boundary conditions additionally specified on it. One of such problems, which leads to a generalized diffusion process, is the object of study in this chapter. It presents the results obtained by the authors concerning the question of construction of an integral representation of a two-parameter Feller semigroup associated with a one-dimensional inhomogeneous diffusion process with a membrane whose properties are described by a nonlocal Feller-Wentzell boundary condition given at it the point of its location.
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