# W-Markov measures, transfer operators, wavelets and multiresolutions Academic Article

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• Overview
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### 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 $\mu_0$ on $(X,\mathcal B_X)$, we then build an associated Markov chain $T_0, T_1, T_2,\ldots$, with these random variables (rv) realized in a suitable probability space $(\Omega,\mathcal F,\mathbb P)$, and each rv taking values in $X$, and with $T_0$ having the probability $\mu_0$ as law. We further show how spectral data for $R$, eg, 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 …

### publication date

• June 24, 2016