Example of an irreducible homogeneous Markov chain, that possesses an invariant measure but is not recurrent

Question

I’m currently studying the topic of Markov chains and how invariant measures are connected to recurrence. I now know that an irreducible homogeneous Markov chain that possesses invariant measures must not also be recurrent, but I can’t find an (easy) expample that illustrate this. Is someone able to provide such an example?

Answer 1

For the deterministic $x \to x+1$ chain on $\mathbb{R}$, the Lebesgue measure is invariant, but this chain is not recurrent. (If you haven't seen the concept of recurrence with uncountable state spaces, disregard this example for now.)

The same dynamics on $\mathbb{Z}$ leave the counting measure invariant, and again this chain is not recurrent.

The gist of this part of the theory is:

• recurrence $\Rightarrow$ an invariant measure exists (but not $\Leftrightarrow$)
• positive recurrence $\Leftrightarrow$ an invariant distribution (i.e. an invariant measure of total measure $1$) exists.