Markov processes either have stationary state or infrared collapse?

Question

The title summarizes my question: Consider a Markov process (discrete or continuous) where the transition-step operator $P$ is independent of time and where mass/probability is conserved for some $L^1$-norm. Do these Markov processes have at least one of the following properties associated to them?

1) A stationary state/measure: a nonzero state which is invariant in time under the Markov process.

2) All measures approach (or even converge to) the zero measure on compact parts of the configuration space.

If we would drop mass-conservation, we could consider the configuration space $\mathbb{R}$ and $P:\mathbb{R} \to \mathbb{R}:$ multiplication by 2 as a counterexample.

Another example to keep in mind is the configuration space $l^1(\mathbb{Z})$ with $P:l^1(\mathbb{Z}) \to l^1(\mathbb{Z})$ the "shift to the right"-operator: $P_{ij}=\delta_{i,j+1}$. This Markov process has no stationary state and since for any state $\mu$ we have $\mu(i,t)=\mu(i-t,0)$, neither do states converge uniformly to the zero state in $L^1$-norm (or any other translation-invariant norm). There is however convergence to $\mu=0$ in $l^1(U \subset \mathbb{Z})$-norm if $U \subset \mathbb{Z}$ is a finite subset of $\mathbb{Z}$.

Answer 1

The Krylov-Bogolyubov theorem seems to be a largely affirmative answer of my question:

For a Feller-continuous Markov-process on a so-called Polish space $X$ with transition measures $(x,t,E\subset X) \mapsto d \mu_{x,t}(E)$ where for at least one point $x^* \in X$ there exists a compact $K\subset X$ such that $d \mu_{x^*,t}(K)>\epsilon$ for some $\epsilon>0$ and all $t>$ some $T>0$, a stationary measure exists.