In many articles, people say "boundedness in probability does imply existence of invariant probability measure", which means$$\sup_{t\ge 0}\mathbb{E}|X_t| \le \infty \quad \Rightarrow \text{there exists probability mesure} ~\mu ~\text{such that} ~\mu P_{t} =\mu.$$$P_{t}(x ,dy)$ is the transition semigroup of the process $(X_t)_{t\ge 0}$.
I don't know why the above statement holds, could you please help me or could you please tell me which books contain its derivation?
Thanks! I have no idea so far.