I read about Markov chains in quite a lot of different resources. However, I can't seem to find a consistent definition of what the requirements are for a Markov chain to have a stationary distribution. (Somewhere, I found that it has to be ergodic and aperiodic, but then again I read that for Markov chain to be ergodic, it must be aperiodic.) And is there any additional requirement so that the stationary distribution is unique? (I found that an irreducible chain has a unique stationary distribution, but how does this connect to the ergodicity?)
I have to admit I'm a bit confused by the various concepts of a chain being ergodic, irreducible, aperiodic, recursive, or fulfilling detailed balance.
Can anyone help me understand all these concepts and definitions better or point me to a good source on this topic?