On connection between random curves, changes of time and regenerative times of random processes

A product space $\Phi\times D$ is considered where $\Phi$ is a set of all continuous non-decreasing functions $\varphi\colon[0,\infty)\to(0,\infty)$, $\varphi(0)=0$, $\varphi(t)\to+\infty$ ($t\to\infty$); $D$ is a set of all right-continuous functions $\xi\colon(0,\infty)\to X$, $X$ is some metric space. Two maps $\Phi\times D\to D$: are defined. The first is the projection $q(\varphi,\xi)=\xi$, and the second is change of time $u(\varphi,\xi)=\xi\circ\varphi$. The following equivalence relation in $D$ is defined:
$$ \zeta_1\sim\xi_2\Leftrightarrow\exists\varphi_1,\varphi_2\in\Phi: \xi_1\circ\varphi_1=\xi_2\circ\varphi_2. $$
Let $M$ is a set of all equivalence classes. Then $L$ is the map $D\to M$: $L\xi_1=L\xi_2\Leftrightarrow\xi_1\sim\xi_2$. $L\xi$ is called the curve corresponding to $\xi$. The following theorem is proved: two random processes with probability measures $P^1$ and $P^2$ on $D$ possess of identical random curves (i.e. $P^1\circ L^{-1}=P^2\circ L^{-1}$) if and only if two random changes of time exist (i.e. two probability measures $Q^1$ and $Q^2$ on $\Phi\times D$) for which $P^1=Q^1\circ q^{-1}$, $P^2=Q^2\circ q^{-1}$) which transform these two processes in a process with a measure $\widetilde{P}$ (i.e. $Q^1\circ u^{-1}=Q^2\circ u^{-1}=\widetilde{P}$). If $(P_x^1)_{x\in X}$ and $(P_x^2)_{x\in X}$ are two families of probability measures for which $P_x^1\circ L^{-1}=P_x^2\circ L^{-1}$ $\forall x\in X$ then for each $x\in X$ corresponding measures $Q^1_x$ and $Q^2_x$ may be found as follows. The set of regenerative times of the family $(\widetilde{P}_x)_{x\in X}$ contains all stopping times which are simultaneously regenerative times of the families $(P^1_x)_{x\in X}$ and $(P^2_x)_{x\in X}$ and have a special first passage time property.