Ergodicity conditions and stationary distributions of a continuous semi-Markov process

A semi-Markov process of a general type on a metric space is considered. The stationary distribution of the process is investigated. A Markov process associated to the semi-Markov one is constructed. The stationary distribution for values of the process at the fixed point of the time scale and for the lengths of intervals of constancy covering this point is expressed in terms of the stationary distribution of the associated Markov process.