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, B X ), we construct an associated Markov chain. Specifically, starting with a choice of R (the transfer operator), and a probability measure µ 0 on (X, B X ), we then build an associated Markov chain T 0 , T 1 , T 2 , . . ., 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 T 0 having the probability µ 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. 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 T 0 = x to T 1 = 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 .

Transfer Operators, Endomorphisms, and Measurable Partitions

“…Under suitable restrictions, in fact, L 2 (R d ) embeds naturally in an L 2 space on the solenoid. We shall refer to the cited literature for details regarding (i)-(iii), but see also [AJL16].…”

“…We finish this section by formulating a result that was proved in [AJL16]. Suppose a positive operator R, acting on measurable function over (X, B, µ), has the properties Rh = h, µR = µ, (4.21) where h is a harmonic function for P and µ is a probability R-invariant measure.…”

“…Indeed, when a transfer operator (R, σ) is given, we show that there is then a naturally induced isometry in the universal Hilbert space H(X), which we show offers a number of applications and results which may be considered to be an infinite-dimensional Perron-Frobenius theory. Our study of transfer operators in L 2 -spaces is motivated by [AJL16,Jor01].…”

“…We will use [Nel69] as a main source. Our analysis of transfer operators in H(X) is motivated by [AJL16,Jor01,Jor04].…”

“…Subsequent research on such questions as symbolic dynamics, spectral theory, endomorphisms in measure spaces, and diffusion processes, further suggest the need for a unifying infinite-dimensional approach. In fact the list of applications is longer than what we already hinted at, and it includes recent joint research involving the second named author with Daniel Alpay, and collaborators; details and citations are included below (see e.g., [AJL13,AJL16,AL13]). This collaborative research also makes use of positive operators and transfer operators in several infinite-dimensional settings, specifically in the study of such stochastic processes as infinite-dimensional Markov transition systems, analysis of Gaussian processes, and in the realization of wavelet multiresolution constructions for a host of probability spaces, and their associated L 2 Hilbert spaces, all of which go beyond the more traditional setting of L 2 (R d ) from wavelet theory.…”