Let $\mathcal{E}$ be a metric space, and suppose that $\mathfrak{B}$ is the Borel field generated by the open sets of $\mathcal{E}$. A stochastic process is defined on $\mathcal{E}$ if a function $x(t,\omega )$$(0 \leqq t < \infty ,\omega \in \Omega )$ and a system of probability measures ${\bf P}_x (x \in \mathcal{E})$ are given such that all ${\bf P}_x $ are defined on the Borel field generated by sets of type (1), and ${\bf P}_x \{ x(0,w) = x\} \equiv 1$. A random variable $\tau (\omega )$ is said to be independent of the future if for every s the $\omega $-set $\{ \tau (\omega ) \leqq s\} $ belongs to the Borel field generated by sets of type (1) with $t \leqq s$. A measurable stochastic process on $\mathcal{E}$ is called a strong Markov homogeneous process if for any $\Gamma _1 , \cdots ,\Gamma _n \in B,0 < t_1 < t_2 < \cdots < t_n $, and any $\tau $ independent of the future (5) is true for almost all $\omega $ such that $\tau (\omega ) < \infty $. Every strong Markov homogeneous process is a Markov...