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On an SVD-based algorithm for identifying meta-stable states of Markov chains. (English) Zbl 1321.65056
Summary: 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 [D. Fritzsche et al., ETNA, Electron. Trans. Numer. Anal. 29, 46–69 (2007; Zbl 1171.15008)], 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.

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A18 Eigenvalues, singular values, and eigenvectors
15B51 Stochastic matrices
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
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