A sufficient condition for the existence of an invariant probability measure for Markov processes

Abstract

In this paper, it is shown that the Foster-Lyapunov criterion is sufficient to ensure the existence of an invariant probability measure for both discrete-and continuous-time Markov processes without any additional hypotheses (such as irreducibility).

Ergodic properties and ergodic decompositions of continuous-time Markov processes

“…where C is a petite set for {X t } t∈R + (since it is petite for R), from which the 'if' part of the result follows. The converse result is Theorem 4.1 of Costa and Dufour (2005a).…”

Φ-entropy inequality and application for SDEs with jumps

“…Moreover, if ∞ 0 ds B(s) = ∞ and A < 0, then W is a compact function and by the Itô formula the solution is non-explosive with EW (X t ) < ∞, t ≥ 0. Thus, according to [4,Theorem 4.1], A < 0 also implies that the associated Markov semigroup has an invariant probability measure. Below we prove that L W is locally bounded such that (3.2) holds.…”

Foster-type criteria for Markov chains on general spaces

“…Theorems 2.1(i) and 2.2(i) in [6] An important feature of our Theorems 4, 5, and 6 for establishing positive Harris recurrence and geometric ergodicity is that they do not require the Markov chain to be ψ-irreducible. Costa and Dufour [1] also showed that the ψ-irreducibility assumption is not needed for the one-step drift condition for positive Harris recurrence. Another subtlety in their drift condition is that it uses extended real-valued test functions -our conditions, and those in [6], simply use real-valued test functions.…”