Question

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.

Answer 1

This is called a "drift condition" in the applied probability literature -- this is used quite often when dealing with MCMC simulations, for example.

In continuous time, what about the good old Ornstein-Uhlenbeck process $dz = -zdt + \sqrt{2}dW$ and generator $L \phi(x) = -x \phi'(x) + \phi^{''}(x)$: the Lyapunov function $V(x) = e^{\alpha |x|}$ works for any $\alpha > 0$.

Answer 2

But why Lyapunov function belongs to the domain of the generator. The Lyapunov function is not bounded.