I am reading about Lyapunov functions for Markov processes, and I am having trouble thinking of examples to keep in mind as I read. If $X_t$ is a continuous-time Markov process with generator $L$, a Lyapunov function is supposed to be a function $V$, in the domain of $L$, with $V \ge 1$ such that $LV \le -aV + b 1_C$, where $a,b$ are constants and $C$ is a "petite" set. It seems that the existence of a Lyapunov function leads to good results on the rate of convergence of $X_t$ to a stationary distribution.
What are some simple examples of processes with explicit Lyapunov functions? Continuous processes would be best. I was trying to think about something like Brownian motion on the circle, but got stuck.