Metastability of finite state Markov chains: a recursive procedure to identify slow variables for model reduction

Abstract

Consider a sequence (η N (t) : t ≥ 0) of continuous-time, irreducible Markov chains evolving on a fixed finite set E, indexed by a parameter N . Denote by R N (η, ξ) the jump rates of the Markov chain η N t , and assume that for any pair of bonds (η, ξ), (η ′ , ξ ′

On the gap between deterministic and probabilistic joint spectral radii for discrete-time linear systems

“…Let us now describe the consequences of this, established in [18]. The following result is an adaptation from Theorems 2.1, 2.7, and 2.12 of [18]. As such an adaptation requires the introduction of several definitions and notations, it is postponed to Appendix A. Theorem 4.5.…”

“…• In [18], the convergence after rescaling in time is given in term of the so-called soft topology, introduced in [16], while in Theorem 4.5 (b) and (c) we state a convergence for the time marginals. The way to get this time marginals convergence instead of the soft one is established in [17, Proposition 2.1].…”

“…• The case of a timescale which is strictly between θ j and θ j+1 is not considered in [18]. In particular, in [18,Condition H1], the total rate of the limit Markov chain is supposed to be nonzero.…”

“…We rigorously handle such a decomposition of Markov processes in different timescales by relying on results from various works by Landim and collaborators, in particular [18], although these works are primarily interested in metastability phenomena, i.e., Markov chains for which the different timescales are all slow, instead of fast as in our case. As a consequence, we prove that we can extract from any sequence of Markov processes with modes in A a subsequence that converges in law to a Markov process associated with a suitable convex combinations of the original matrices in A.…”

Metastable Markov chains

“…The results of this subsection are taken from Section 4 of [95]. We refer to [93] for an application.…”