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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)

See also Markov Process




© 1996-9 Eric W. Weisstein
1999-05-26