{
  "id": "0903.3267",
  "version": "v1",
  "published": "2009-03-19T03:00:34.000Z",
  "updated": "2009-03-19T03:00:34.000Z",
  "title": "Spectral Theory of Discrete Processes",
  "authors": [
    "Palle E. T. Jorgensen",
    "Myung-Sin Song"
  ],
  "comment": "34 pages with figures removed, for the full version with all the figures please go to http://www.siue.edu/~msong/Research/spectrum.pdf",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-19T03:00:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C40",
      "47S50",
      "68U10",
      "94A08"
    ],
    "keywords": [
      "spectral theory",
      "discrete processes",
      "random walk",
      "von neumanns spectral theorem",
      "stochastic processes"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.2478/s11534-009-0119-4",
      "journal": "Open Physics",
      "year": 2010,
      "month": "Jun",
      "volume": 8,
      "number": 3,
      "pages": 340
    },
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010OPhy....8..340J"
    }
  }
}