W-Markov measures, transfer operators, wavelets and multiresolutions

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 $T_0,T_1,T_2,\dots$, with these random variables (r.v) realized in a suitable probability space $(\Omega ,\mathcal{F},\mathbb{P})$, and each r.v. taking values in $X$, and with $T_0$ having the probability ${\mu }_0$ 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.

References in corpus (1)

• Contributions of Issai Schur to Analysis

Cited by in corpus (2)

• Infinite-dimensional transfer operators, endomorphisms, and measurable partitions
• Separable representations of higher-rank graphs