Abstract
In the setting of stochastic dynamical systems that eventually go extinct, the quasi-stationary distributions are useful to understand the long-term behavior of a system before evanescence. For a broad class of applicable continuous-time Markov processes on countably infinite state spaces, known as reaction networks, we introduce the inferred notion of absorbing and endorsed sets, and obtain sufficient conditions for the existence and uniqueness of a quasi-stationary distribution within each such endorsed set. In particular, we obtain sufficient conditions for the existence of a globally attracting quasi-stationary distribution in the space of probability measures on the set of endorsed states. Furthermore, under these conditions, the convergence from any initial distribution to the quasi-stationary distribution is exponential in the total variation norm.
Highlights
We may think of reaction networks in generality as a natural framework for representing systems of transformational interactions of entities [48]
The set of entities may in principle be of any nature, and specifying not just which ones interact and quantifying how frequent they interact, we obtain the dynamical system of a reaction network
We will prove as the main result in Theorem 5.1 and Corollary 5.2 sufficient conditions for the existence of a unique globally attracting quasi-stationary distribution (QSD) in the space of probability distributions on E, equipped with the total variation norm, · TV
Summary
We may think of reaction networks in generality as a natural framework for representing systems of transformational interactions of entities [48]. We will prove as the main result in Theorem 5.1 and Corollary 5.2 sufficient conditions for the existence of a unique globally attracting QSD in the space of probability distributions on E, equipped with the total variation norm, · TV. Recall that this norm may be defined as [39]. This approach has been applied to a particular case of multidimensional birth-death processes, giving sufficient conditions, in terms of the parameters of the process, for the existence and uniqueness of a QSD We extend this result, not just to a larger set of parameter values in the birth-death process case, but to the much broader class of stochastic processes known as stochastic reaction networks.
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