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We use analytical methods to construct the two-parameter Feller semigroupassociated with a Markov process on a line with a moving membrane such that atthe points on both sides of the membrane it coincides with the ordinarydiffusion processes given there and its behavior after reaching the membrane isdescribed by a kind of nonlocal Feller-Wentzell conjugation condition.

One-dimensional diffusion processes with moving membrane: partial reflection in combination with jump-like exit of process from membrane