## On an SVD-based algorithm for identifying meta-stable states of Markov chains

Ryan M. Tifenbach

### Abstract

A Markov chain is a sequence of random variables $X = \{x_{t}\}$ that take on values in a state space $\mathcal{S}$. A meta-stable state with respect to $X$ is a collection of states $\mathcal{E} \subseteq \mathcal{S}$ such that transitions of the form $x_{t} \in \mathcal{E}$ and $x_{t+1} \notin \mathcal{E}$ are exceedingly rare. In Fritzsche et al. [Electron. Trans. Numer. Anal., 29 (2008), pp. 46–69], an algorithm is presented that attempts to construct the meta-stable states of a given Markov chain. We supplement the discussion contained therein concerning the two main results.

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### Key words

Markov chains; conformation dynamics; singular value decomposition

### AMS subject classifications

15A18, 15A51, 60J10, 60J20, 65F15

### Links to the cited ETNA articles

 [6] Vol. 29 (2007-2008), pp. 46-69 David Fritzsche, Volker Mehrmann, Daniel B. Szyld, and Elena Virnik: An SVD approach to identifying metastable states of Markov chains