On Conditional Markov Processes

In this paper a pair of random processes $X_t$, $Y_t$, which conjunctly form the Markov process $Z_t$ is considered. The conditional distribution of the process $Y_t$ for the condition of a known realization of the process $X_t$ during some time interval is examined. E. B. Dynkin has proposed that if $X_t$ is a Markov process, the conditional distribution of $Y_t$ will satisfy a functional equation similar to the known Kolmogorov–Chapman equation. The author has proved this proposition, but details of the proof are omitted here.