I am really confused for the continuous time Markov process when they use the notation such as $E_x[Z]$ or $E_{\mu}[Z]$, and even more for $E_x[Z]|_{x=Y_t}$.
The Markov property expressed for the canonical process state as the following: For any $t \geqslant 0$ and $ \mathcal{Y}_{\infty}^0$-measurable randome variable $Z \geqslant 0$ on $S^{[0,\infty)}$, we have $E_{\nu}[Z \cdot \theta_t]=E_{Y_t}[Z]:=E_x[Z]|_{x=Y_t}$
I personally think that the $E_x[Z]$ is the expectation for the probability measure $P_x$ which is $P_x:=\delta(x)$ and $x \in S$ where S is the state space. What does it mean for $E_x[Z]|_{x=Y_t}$？ Since $Y_t$ is a process. Any one can give me more detailed definition? I am not sure $\delta(Y_t=x)$ mean here?g