Semi-Markov processes on a general state space: α-theory and quasi-stationarity

Abstract

By amalgamating the approaches of Tweedie (1974) and Nummelin (1977), an a-theory is developed for general semi-Markov processes. It is shown that x-transient, 2-recurrent and 2-positive recurrent processes can be denned, with properties analogous to those for transient, recurrent and positive recurrent processes. Limit theorems for a-positive recurrent processes follow by transforming to the probabilistic case, as in the above references: these then give results on the existence and form of quasistationary distributions, extending those of Tweedie (1975) and Nummelin (1976).

Quasi-stationary distributions for randomly perturbed dynamical systems

“…When P(X 1 ∈ ·) = µ(·), λ is the probability of not being absorbed in the next time step. The existence of QSDs has been studied extensively [Darroch and Seneta (1965), Seneta and Vere-Jones (1966), Tweedie (1974), Barbour (1976), Nummelin and Arjas (1976), Arjas, Nummelin and Tweedie (1980), Kijima (1992), Ferrari et al (1995), Chan (1998), Lasserre and Pearce (2001), Gosselin (2001), Coolen-Schrijner and van Doorn (2006), Buckley and Pollett (2010)]. Högnäs (1997), Klebaner, Lazar and Zeitouni (1998), Ramanan and Zeitouni (1999) studied weak* limit points µ of QSDs µ ε as ε → 0 for maps of the interval, that is, M = [0, 1] and M 0 = {0}.…”

Quasi-Stationary Distributions

“…The nineteen sixties and seventies saw further developments in the theory of quasi-stationary distributions for countable-state Markov chains (for example Flaspohler [59], Tweedie [143]), as well as generalizations to semi-Markov processes (Arjas et al [4], Cheong [28,29], Flaspohler and Holmes [60], Nummelin [110]) and Markov chains on a general state space (Tweedie [144]), and detailed results for generic models, for example, birth-death processes (Cavender [23], Good [66], Kesten [78]), random walks (Daley [36], Pakes [115], Seneta [131]), queueing systems (Kyprianou [92,93,94]) and branching processes (Buiculescu [20], Evans [52], Green [64], Seneta and Vere-Jones [137]).…”

Cheeger inequalities for absorbing Markov chains

“…In the finite-state case, the authors in [2] show that the empirical profile of the unique invariant measure of the Fleming-Viot process with N particles converges as N → ∞ to the unique quasistationary distribution of the one-particle motion. Other settings that have been considered include semi-Markov processes [1], [5], [17], continuous-state processes [34], and settings where P may be reducible [36], [37]. Applications of substochastic Markov chains include biological, environmental, and financial models, where frequently a path terminates when it enters a death or valueless state.…”

“…Our particular focus in this paper is on conductance and metastability in Markov processes. The conductance of a Markov chain (see Definition 2) is the minimum conditional probability of leaving a set B in one step of the process, minimised over all sets B ⊂ X with probability no greater than 1 2 . The conductance has many other names within the literature, such as the Cheeger constant [4], the isoperimetric constant [26], and the bottleneck ratio [27].…”

“…The conductance has many other names within the literature, such as the Cheeger constant [4], the isoperimetric constant [26], and the bottleneck ratio [27]. The metastability of a Markov chain (see Definition 2) is the maximum conditional probability of remaining in a set B for one step of the process, maximised over all sets B ⊂ X with probability no greater than 1 2 . The study of metastability has foundations in both Markov chain theory [26] and in the study of dynamical systems, where it is also called almost-invariance [10], [19].…”