Approximating selected non-dominant timescales by Markov state models

We consider time-continuous, reversible Markov processes on large or continuous state space. For a practical analysis of such processes it is often necessary to construct low dimensional approximations, like Markov State Models (MSM). MSM have been used for this purpose in several applications, particularly in molecular dynamics; see [16] for an example. One of the main goals of MSMs is the correct approximation of slow processes in the system. Recently, it was possible to understand under which conditions a MSM inherits the most dominant timescales of the original Markov process [7, 6]. However, all rigorous statements known have yet been concerned with the approximation of the absolutely slowest processes in the system, i.e., its dominant timescales. In this article, we will show that it is also possible to design MSMs to reproduce selected non-dominant timescales and which approximation quality can be achieved.

Keywords

Markov process, metastability, multiscale, eigenvalue problem, transfer operator, eigenvalue error, Markov state models, committor, Galerkin approximation

2010 Mathematics Subject Classification

00A69, 60J35

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Published 9 April 2012