Spectral theory of discrete processes

Abstract

We offer a spectral analysis for a class of transfer operators. These transfer operators arise for a wide range of stochastic processes, ranging from random walks on infinite graphs to the processes that govern signals and recursive wavelet algorithms; even spectral theory for fractal measures. In each case, there is an associated class of harmonic functions which we study. And in addition, we study three questions in depth

In specific applications, and for a specific stochastic process, how do we realize the transfer operator T as an operator in a suitable Hilbert space? And how to spectral analyze T once the right Hilbert space H has been selected? Finally we characterize the stochastic processes that are governed by a single transfer operator.

In our applications, the particular stochastic process will live on an infinite path-space which is realized in turn on a state space S. In the case of random walk on graphs G, S will be the set of vertices of G. The Hilbert space H on which the transfer operator T acts will then be an L ² space on S, or a Hilbert space defined from an energy-quadratic form.

This circle of problems is both interesting and non-trivial as it turns out that T may often be an unbounded linear operator in H; but even if it is bounded, it is a non-normal operator, so its spectral theory is not amenable to an analysis with the use of von Neumann’s spectral theorem. While we offer a number of applications, we believe that our spectral analysis will have intrinsic interest for the theory of operators in Hilbert space.