# 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?

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.

• Yeah, but what does $P_t$ mean? Is it the t'th step transition matric or the transition matrix at time t? Or are these two things the same? – Kaish Nov 22 '12 at 15:10
• $T^n$ is the $n$-th step transition matrix, i.e. the probability of finding the walker in state $y$ after $n$ steps, given it has started in state $x$ is $(y,T^n x)$. Similarly, the probability of being in state $y$ after time $t$ and starting at $x$ is $(y,P_t x)$. – Sven Stodtmann Nov 22 '12 at 15:53

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 $t_0$.

• +1. Thank you--this question has long needed a useful, relevant answer. – whuber Sep 4 '15 at 13:27