W-Markov measures, transfer operators, wavelets and multiresolutions

2016/06/24 by Alpay, Daniel, Jorgensen, Palle, Lewkowicz, Izchak
#37C30 #46L55 #47B65 #60J05 #60J10 #65T60 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1606.07692

Abstract

In a general setting we solve the following inverse problem: Given a positive operators R, acting on measurable functions on a fixed measure space (X,\mathcal B_X), we construct an associated Markov chain. Specifically, starting with a choice of R (the transfer operator), and a probability measure μ₀ on (X, \mathcal B_X), we then build an associated Markov chain T₀, T₁, T₂,…, with these random variables (r.v) realized in a suitable probability space (Ω,\mathcal F, \mathbb P), and each r.v. taking values in X, and with T₀ having the probability μ₀ as law. We further show how spectral data for R, e.g., the presence of R-harmonic functions, propagate to the Markov chain. Conversely, in a general setting, we show that every Markov chain is determined by its transfer operator. In a range of examples we put this correspondence into practical terms: (i) iterated function systems (IFS), (ii) wavelet multiresolution constructions, and (iii) IFSs with random control.

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