Interpretation of "scale function" in Foster-Lyapunov drift condition I'm reading about Markov chains and I'm starting to bump into these drift conditions, and their relationship with a chain's ergodic properties. The drift condition is that there exists a "scale function" $V: X \to [1,\infty)$, a small set $\mathcal{C}$, and constants $\beta \in (0,1)$, $b < \infty$ such that
$$
PV(x) = \int P(x,dy)V(y) \le \beta V(x) + b\mathbb{1}_{\mathcal{C}}(x).
$$
I'm wondering if anyone can give me some direction on how to interpret this $V$ function, especially in the context of analyzing the convergence of MCMC algorithms, and especially if this interpretation can be supported by some results. Right now I imagine that $V$ is some sort of distance function describing how far away from the "good spot" you currently are, as the expected value of $V$ one step in the future decreases, unless of course you are already in this good (small?) set.
 A: Good question, and personally, I often find myself re-explaining the concept after the umpteenth time I think I have understood it. 
Your intuition is pretty much right, in that, $V$ serves as somewhat of a "distance" metric, but not quite. One common way of interpreting $V$ is by thinking of it as a potential energy surface, as describe by Jones and Hobert (2001). They write

It is useful to think of $V$ as a potential energy surface. When the
  Lypanuov condition holds, the chain tends to “drift” toward states of
  lower energy in expectation. In this context, V is called an energy
  function.

Often the small set is described as the set on which $V$ takes small values (this is often called the associated minorization condition). So in essence, when the chain is not in the "small" set, $V$ takes large values and thus has large energy, and so the drift condition forces the average update on the drift function to be smaller, forcing it to go towards the small set. When the chain is in the small set, the term $b$ allows for it to move outside the small set freely. 
The trick is to ensure $V$ takes low values in area of reasonable probability under $\pi$, the stationary distribution. A recent paper by Qin and Hobert (2018), explain that if $V$ is "centered" in the sense that it takes small values in areas of high $\pi$ probability, this leads to a "stable" drift function, and yields good bounds on the rate of convergence.
All together, this means that the Markov chain moves towards the places of high probability when it can, and when it is already in an area of high probability, it is free to jump and move about freely. Of course, this then says that the Markov chain explores the state space well, and hence convergence at a reasonable rate.

I have skipped the discussion on the minorization condition, and the concept of coupling, which is also important to this discussion, but you did not ask about that. If you are interested, the Jones and Hobert (2001) lays it out pretty well.
