Let $(Z_t)_{t\geq0}$ be a $d$-dimensional stochastic process with invariant measure $P$ and $Z\sim P$. Let $(P_t)_{t\geq0}$ denote the transition semigroup of the process, i.e. in particular for $h:\mathbb{R}^d \to \mathbb{R}, x\in \mathbb{R}^d$ $$ P_th(x) = \mathbb{E}_P[h(Z_{t,x})], $$ where $Z_{t,x}$ denotes the state of the process at time $t$ when starting at $x$. I've come across the equation $$ \mathbb{E}_P[h(Z)] = \mathbb{E}_P[(P_th)(Z)]. $$ Unfortunately, that is not obvious to me. Can someone explain where that comes from? Thanks a lot!
Edit: Is this obvious, because $P$ is the invariant measure of the process, so $Z_{t,x} \sim P$ regardless of the starting point $x$? Therefore $Z_{t, Z}$ and $Z$ follow the same distribution and $$ \mathbb{E}_P[(P_th)(Z)] = \mathbb{E}_P[\mathbb{E}_P[h(Z_{t, Z})]] = \mathbb{E}_P[\mathbb{E}_P[h(Z)]] = \mathbb{E}_P[h(Z)]? $$
This seems to be indeed obvious, and my problem was that I did not find a clear definition of the term "invariant measure". I found that now so I'm going to answer myself. (Maybe someone can confirm that this is correct.)
Apparently, an invariant measure is a distribution that does not change under the flow of the stochastic process, i.e. if $Z_0\sim P$, then $Z_t\sim P$ for all $t\geq0$. Then by definition, $Z_{t, Z}$ and $Z$ follow the same distribution and the equation in the Edit section should be correct.