On one-dimensional diffusion processes with moving membranes

. Using the method of the classical potential theory, we construct the two-parameter Feller semigroup associated, on the given interval of the real line, with the Markov process such that it is a result of pasting together, at some point of the interval, two ordinary diffusion processes given in sub-domains of this interval. It is assumed that the position on the line of boundary points of these sub-domains depends on the time variable. In addition, some variants of the general nonlocal boundary condition of Feller-Wentzell’s type are given in these points. The resulting process can serve as a one-dimensional mathematical model of the physical phenomenon of diffusion in media with moving membranes.