Most books show Ito diffusions satisfy Markov property, that is, $E[f(X_{t+h})\mid F_t]=E^{X_t}[f(X_h)]$. But I was wondering whether it's true that $E[f(X_{t+h})\mid X_t]=E^{X_t}[f(X_h)]$.
In this post, it seems that stationarity of transition of the processes implies $E[f(X_{t+h})\mid X_t] = E^{X_t}[f(X_h)]$. If it's true, my question is solved. But I don't know why $E^{X_t}[f(X_h)]$ is $X_t$ measurable and how to connect it to the transition probability.