W-Markov measures, transfer operators, wavelets and multiresolutions

In a general setting we solve the following inverse problem: Given a positive operator R, acting on measurable functions on a fixed measure space (X, 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, 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, F, 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.” Our setting for IFSs is general as well: a fixed measure space (X, B_X) and a system of mappings τ_i, each acting in (X, B_X), and each assigned a probability, say p_i which may or may not be a function of x. For standard IFSs, the p_i ’s are constant, but for wavelet constructions, we have functions p_i(x) reflecting the multi-band filters which make up the wavelet algorithm at hand. The sets τ_i(X) partition X, but they may have overlap, or not. For IFSs with random control, we show how the setting of transfer operators translates into explicit Markov moves: Starting with a point x ∈ X, the Markov move to the next point is in two steps, combined yielding the move from T₀ = x to T₁ = y, and more generally from T_n to T_n+1. The initial point x will first move to one of the sets τ_i (X) with probability p_i, and once there, it will “choose” a definite position y (within τ_i(X)), now governed by a fixed law (a given probability distribution). For Markov chains, the law is the same in each move from T_n to T_n+1.