Ergodic properties of Markov processes
In these notes we discuss Markov processes, in particular stochastic differential equations (SDE) and develop some tools to analyze their long-time behavior. There are several ways to analyze such properties, and our point of view will be to use systematically Liapunov functions which allow a nice characterization of the ergodic properties. In this we follow, at least in spirit, the excellent book of Meyn and Tweedie . In general a Liapunov function W is a positive function which grows at infinity and satisfies an inequality involving the generator of the Markov process L: roughly speaking we have the implications (α and β are positive constants)
L Rey-Bellet. "Ergodic properties of Markov processes" Open Quantum Systems II 1881 (2006): 1-39.
Available at: http://works.bepress.com/luc_rey_bellet/16