Transfer Operators, Induced Probability Spaces, and Random Walk Models

TL;DR

This paper develops a framework connecting transfer operators, harmonic functions, and probability measures to realize random walk models in dynamical systems, providing spectral analysis and path-space constructions.

Contribution

It introduces a general method to construct random walk models from transfer operators using harmonic functions and invariant measures, with explicit path-space realizations and spectral characterizations.

Findings

Constructs path-space probability measures from transfer operators.

Provides spectral characterization of the induced automorphisms.

Connects transfer operators with Lax-Phillips scattering theory.

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 $\left(h,\lambda\right)$, where $h$ is an $R$-harmonic function on $X$, and $\lambda$ 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 $\mathbb{P}$ 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 $\mathbb{P}$ quasi-invariant with respect to a shift automorphism. The latter takes the form of explicit multi-resolutions in $L^{2}$ of $\mathbb{P}$ in the sense of Lax-Phillips scattering theory.