Yet another look at Harris' ergodic theorem for Markov chains

Setting and main result

Throughout this note, we fix a measurable space $\mathbf{X}$ and a Markov transition kernel $\mathcal{P}(x,\cdot)$ on $\mathbf{X}$. We will use the notation $\mathcal{P}$ for the operators defined as usual on both the set of bounded measurable functions and the set of measures of finite mass by

Hence we are using $\mathcal{P}$ both to denote the action on functions and its duel action on measure. Note that $\mathcal{P}$ extends trivially to measurable functions $\varphi\colon\mathbf{X}\to[0,+\infty]$. We first assume that $\mathcal{P}$ satisfies the following geometric drift condition:

There exists a function $V:\mathbf{X}\rightarrow[0,\infty)$ and constants $K\geq 0$ and $\gamma\in(0,1)$ such that