Lyapunov Functions and Classification of Markov Chains
Denis Denisov, Dmitry Korshunov, Vitali Wachtel · Cambridge University Press eBooks · 2025
In Chapter 2 we introduce a classification of Markov chains with asymptotically zero drift, which relies on relations between the drift and the second moment of jumps, with many improvements on the results known in the literature. Additional assumptions are expressed in terms of truncated moments of higher orders and tail probabilities of jumps. Another, more important, contrast with previous results on recurrence/transience is the fact that we do not use concrete Lyapunov test functions (quadratic or similar). Instead, we construct an abstract Lyapunov function which is motivated by the harmonic function of a diffusion process with the same drift and diffusion coefficient.