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.

  • $\begingroup$ Just for the fun, let me bring here my two "centimes". A set is of masculin gender in french (un ensemble) so it would be more something like 'a "petit" set', but I agree that it doesn't sound as good as 'a petite'" set'. Interesting question by the way. $\endgroup$ – The Bridge Mar 23 '11 at 7:19

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 $\alpha > 0$.

  • $\begingroup$ Thanks, this looks like a good example. If I take $\alpha = 1$, $a=1$, I guess my petite set is $[-2,2]$. $\endgroup$ – Nate Eldredge Mar 23 '11 at 16:04
  • $\begingroup$ What about $d z = - U'(z) d t + \sqrt{2} d W$ where $U$ is confining potential? $\endgroup$ – megaproba Jul 4 '17 at 9:36

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


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