Foster's theorem states that the Markov chain is positive recurrent if and only if there exists a Lyapunov function V : S → R {\displaystyle V:S\to \mathbb {R} }.

As long as P. N i=1 λi < 1, {Xn} is positive recurrent. Proof. By Foster's theorem, define the Lyapunov function: L(x) = 1. 2.

Foster-Liapunov Criterion. In what follows, X = (Xn). ∞ n=0 is a Markov chain with countable state space S and transition probability matrix P = {p(x, y)}x,y ...

Let Φ = {Φn,n ≥ 0} be a time-homogeneous Markov chain on a state space X. (Assume that Φ is phi-irreducible and aperiodic, for simplicity.).

This paper surveys such powerful stochastic Lyapunov function methods for general state space Markov processes. Under the hypothesis of p-irreducibility, which ...

Originally, Lyapunov functions were developed by A. Lyapunov in 1899 for the study of sta- bility of dynamical systems described by ODE's, mainly motivated ...

In Part I we developed stability concepts for discrete chains, together with Foster–Lyapunov criteria for them to hold. Part II was devoted to developing ...

The main objective in this thesis is to obtain a unified approach to the stability classification of continuous-time Markov chains defined on discrete state ...

3 сент. 2024 г. · In this paper we investigate the ergodicity in total variation of the process X_t related to some integro-differential operator with unbounded coefficients.

In this paper we develop criteria for these forms of stability for continuous-parameter Markovian processes on general state spaces, based on Foster-Lyapunov ...