Smith's Markov Process Theorem

Consider

$\begin{displaymath} P_2(y_1,t\vert y_3,t_3) = \int P_2(y_1,t_1\vert y_2,t_1)P_3(y_1,t_1;y_2,t_2\vert y_3,t_3)\,dy_2. \end{displaymath}$

(1)

If the probability distribution is governed by a Markov Process, then

$\displaystyle P_3(y_1,t_1;y_2,t_2\vert y_3,t_3)$	$\textstyle =$	$\displaystyle P_2(y_2,t_2\vert y_3,t_3)$
	$\textstyle =$	$\displaystyle P_2(y_2\vert y_3,t_3-t_2).$	(2)

Assuming no time dependence, so $t_1\equiv 0$ ,

$\begin{displaymath} P_2(y_1\vert y_3,t_3)=\int P_2(y_1\vert y_2,t_2)P_2(y_2\vert y_3,t_3-t_2)\,dy_2. \end{displaymath}$

(3)