A time homogeneous Markov process on Ω with semi-group $P_t$ is said to stationary w.r.t a distribution $π$ if
$$∫_{Ω}P_tf(x)dπ(x)=∫_{Ω}f(x)dπ(x), \text{for $f$ bounded measurable}.$$
and reversible if
$$∫_{\Omega}f(x)P_tg(x)dπ(x)=∫_{Ω}g(x)P_tf(x)dπ(x), \text{for $f,g$ bounded measurable.}$$
These properties are frequently be translated into words as : $\textbf{Stationary}$ once the process reaches $π$ it never leaves, $\textbf{Reversible}$ people say it is like "if you watched a movie backwards it would look the same as if you watched it forwards in time".
I completely understand the meaning of $\textbf{Stationary}$ . However I dont really understand the movie analogy, could anyone make this more concrete, i.e. could I run a simulation and observe this analogy?