Given a continuous-time continuous-state Markov process:
A Markov process $X$ is ergodic if there exists a unique invariant probability distribution $\mu$ and, for any state $x$, the transition probabilities $P(X_t ∈ ·|X_0 = x)$ converge to that distribution in total variation.
and
A Markov process $X$ having invariant (probability) measure $\mu$ is a stationary process if $X_0 \sim \mu$.
I am confused because I thought that ergodicity implied stationarity, but according to these two definitions, it seems that the process $X$ with $X_0 = x$, a point mass, would still be ergodic but not stationary.
edit: I am interested in whether Birkhoff ergodic theorem applies to $X$ with $X_0=x$.