Diffusion processes with delay on ends of a segment

A continuous semi-Markov process with a segment as a range of values is considered. This process is being transformed into a diffusion process inside the segment, i.e., up to the first hitting time on the boundary of the segment and any time leaving the boundary. Some conditions in terms of a semi-Markov transition generating function on the boundary for such a process to exist are derived. A method of imbedded alternating renewal processes is applied to find a stationary distribution of the process.