On extensions of a Markov process

Let $D$ be an open set in a compact metric space $E$. A Markov process $X$ in $E$ is called an extension of a process $X^0$ given in $D$, if the part of $X$ on $D$ is equivalent to $X^0$. In this paper characteristics are introduced which describe extensions $X$ of a process $X^0$. An analogous problem was recently treated by Motoo [4]. We investigate the problem by other methods and under more general conditions.