Definition of stationary distribution in continuous time markov chains I found the following definition:
"A probabilitly distribution $\pi = \{\pi_x\}_{x \in S}$ on the state space $S$ is called a stationary distribution for the Markov chain if for every $t > 0$,
$$
\pi^T P_t = \pi^T
$$
What does $P_t$ mean? I thought it was the t'th step matrix of the transition matrix P but then this would be for discrete time markov chains and not continuous, right?
Oh wait, is it the transition matrix at time t?
 A: Please read the correct answer from the other post, as mentioned there, this answer incorrectly assumes a rate matrix instead of a transition Matrix.
You can always get a continuous time version of a discrete one by simply "Poissonizing" it. For example, if you have a discrete time Markov chain with transition matrix $T$ you get a continuous time version by considering
$$P_t = \sum_{n\geq 0} \frac{t^n}{n!}\exp(-t)T^n $$
Hence the above definition makes sense in the context of continuous time Markov chains.
A: I am answering rather than commenting due to lack of reputation:
Sven, your claim is incorrect: in your expression, T must be an infinitesimal rate matrix whose rows sum to 0, not a transition matrix whose rows sum to 1.
And now actually answering:
With CTMCs, different things happen to those who wait longer. $P_t$ denotes a transition matrix between observations at time $t_0$ and time $t_0 + t$. When $t$ goes to $0$, it approaches the identity matrix, and when $t$ goes to infinity, it approaches a matrix where every row in $\pi$. Those claim may require regularity conditions such as irreducibility. My entire answer also assumes the process is time-homogenous, i.e. $P_t$ doesn't depend on $t_0$.
