Spectral Analysis of Markov Jump Processes with Rare Transitions: A Graph-Algorithmic Approach

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2017

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Abstract

Parameter-dependent Markov jump processes with exponentially small transition

rates arise in modeling complex systems in physics, chemistry, and biology.

Long-term dynamics of these processes are largely governed by the spectral properties

of their generators. We propose a constructive graph-algorithmic approach to

computing the asymptotic estimates of eigenvalues and eigenvectors of the generator

matrix. In particular, we introduce the concepts of the hierarchy of Typical Transition

Graphs (T-graphs) and the associated sequence of Characteristic Timescales.

The hierarchy of T-graphs can be viewed as a unication of Wentzell's hierarchy of

optimal W-graphs and Friedlin's hierarchy of Markov chains. T-graphs are capable

of describing typical escapes from metastable classes as well as cyclic behaviors

within metastable classes, for both reversible and irreversible processes, with or

without symmetry. Moreover, the hierarchy of T-graphs can be used to construct

asymptotic estimates of eigenvalues and eigenvectors simultaneously. We apply the

proposed approach to investigate the biased random walk of a molecular motor and

conduct zero-temperature asymptotic analysis of the LJ75 network.

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