Construction of Non-Homogeneous Markov Processes by Means of a Random Substitution of Time

It is proved that a continuous single-dimensional Markov process $y(t)$ with wide restrictions can be obtained from the Wiener process $x(t)$ in the following form: $y(t)=\psi[x(\tau_t),t]$, where $\psi(x,t)$ is a continuous function, monotonic in $x$ for a given $t$, and $\tau _t $ is a non-decreasing random function of $t$ (Theorem 1).
Conditions are given which should be met by the Markov process $x(t)$ in abstract space and the random function $\tau_t$ so that the process $y(t)=x(\tau_t)$ will also be a Markov process (Theorem 2).