On Markov diffusion processes with delayed reflection from boundaries of a segment

A continuous semi-Markov process with values in a closed interval is considered. This process coincides with a Markov diffusion process inside the interval. Thus, violation of the Markov property is only possible at the boundary of the interval. We prove a sufficient condition under which a semi-Markov process is Markov. We show that, in addition to Markov processes with instantaneous reflection from the boundary of the interval. there exists a class of Markov processes with delayed reflection from the boundary. Such a process has a positive average measure of time at which its trajectory belongs to the boundaries. This gives a different proof of a similar result by Gikhman and Skorokhod of 1968. Bibliography: 5 titles.