In probability theory, when studying the convergence of a stochastic process to equilibrium minorisation conditions can be exploited (see for instance Assumption 2.1. of https://www.sciencedirect.com/science/article/pii/S0304414902001503?ref=pdf_download&fr=RR-2&rr=7e1ea215b82bb2e7).
What is the intuitive reason that such a condition can provide convergence to equilibrium, I have heard in passing that it is related to coupling techniques but can not see how.
Not sure if this helps for the above specific paper, but for the Discrete case the reasoning is much easier to explain.
For any kernel $K$ acting from $\mathcal{X} \to \mathcal{Y}$, we can decompose it as convex combination of two Kernels as $$ K = \alpha 1 \pi^T + (1 − \alpha)M $$ where $\alpha = \sum_{y \in \mathcal{Y}} \min_{x \in \mathcal{X}} K(y|x) $ is the Doeblin coefficient of Kernel $K$ and $\pi(y) = \min_{x \in \mathcal{X}} K(y|x)/\alpha $.
Now, whenever a measure $\mu$ is acted upon by a kernel $K$, there is essentially an 'erasure of the $\alpha$-th fraction of information' in $\mu$ due to the kernel $1 \pi^T $ in the above decomposition. This idea is basically at the heart of many convergence results.