Transfer Operators, Induced Probability Spaces, and Random Walk Models

Abstract

We study a family of discrete-time random-walk models. The starting point is a fixed generalized transfer operator R subject to a set of axioms, and a given endomorphism in a compact Hausdorff space X. Our setup includes a host of models from applied dynamical systems, and it leads to general path-space probability realizations of the initial transfer operator. The analytic data in our construction is a pair (h, λ), where h is an R-harmonic function on X, and λ is a given positive measure on X subject to a certain invariance condition defined from R. With this we show that there are then discrete-time random-walk realizations in explicit path-space models; each associated to a probability measures P on path-space, in such a way that the initial data allows for spectral characterization: The initial endomorphism in X lifts to an automorphism in path-space with the probability measure P quasi-invariant with respect to a shift automorphism. The latter takes the form of explicit multi-resolutions in L 2 of P in the sense of Lax-Phillips scattering theory.