ON MARKOV CHAINS AND THE SPECTRA OF THE CORRESPONDING FROBENIUS–PERRON OPERATORS

Abstract

The paper presents some notes on the theoretical background of recent computational approaches to identify the dynamical behavior of Markov chains by solving an eigenvalue problem of the Frobenius–Perron operator. We show that for each subset of its peripheral spectrum, the corresponding generalized eigenspace and the geometric eigenspace coincide. Furthermore, the peripheral spectrum forms a cyclic set. The elements of this set stand in a one-to-one relationship to cyclic and invariant behavior of the underlying stochastic process. If the dynamics of the chain is perturbed continuously, invariant sets of the unperturbed chain possibly merge and become almost invariant. We give a precise continuity condition under which the merging of two invariant sets and the appearance of almost invariant sets can be observed and follow how the corresponding eigenmeasure leaves the eigenspace associated to 1. Finally, as an example we present the appearance of almost invariance for a Markov chain on a compact interval.