Stationary MC distribution

If you can find a distribution in a Markov chain where $p_{t} = p \Rightarrow p_{t + t} = p$ then each of the following state distributions will be $p$ . Thus, once a Markov chain gets into state distribution $p$ , then it is a Stationary time series.

Formally the distribution must satisfy $p = K p$ or $p (y) = \int_{x} K (x, y) p (x) d x$ So $p$ must be a (unit) eigenvector or eigen function with an eigenvalue of 1.