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 $\mathcal{H}$ 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 $\mathcal{H}$ 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 $\mathcal{H}$; 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.