{
  "id": "2311.08564",
  "version": "v1",
  "published": "2023-11-14T21:55:12.000Z",
  "updated": "2023-11-14T21:55:12.000Z",
  "title": "Probabilistic Representations of Ordered Exponentials: Vector-Valued Schrödinger Semigroups and the Combinatorics of Anderson Localization",
  "authors": [
    "Pierre Yves Gaudreau Lamarre"
  ],
  "comment": "54 pages, 10 figures; comments welcome",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We provide two applications of an elementary (yet seemingly unknown) probabilistic representation of matrix ordered exponentials, which generalizes the Feynman-Kac formula in finite dimensions and the change of measure formula between two continuous-time Markov processes on a finite state space. Our first and main application consists of a new Feynman-Kac formula for a class of vector-valued Schr\\\"odinger operators on the line, which is driven by two sources of randomness: The usual Brownian motion, and a continuous-time Markov process on a finite state space. An important feature of these formulas -- which is at the core of our motivation -- is that they enable the calculation of the joint moments of the semigroup kernels when the matrix potential function contains a continuous Gaussian noise. In particular, our moment formulas shed new light on what the joint moments of the Feynman-Kac kernels of the multivariate stochastic Airy operators of Bloemendal and Vir\\'ag (Ann. Probab., 44(4):2726--2769, 2016.) should be; we state a precise conjecture to that effect, which we pursue in a forthcoming paper. Our second application consists of Feynman-Kac formulas for the expected square modulus $\\mathbf E\\big[|\\Psi(t,x)|^2\\big]$ of the solutions of the Schr\\\"odinger equation $\\partial_t\\Psi=-\\mathsf i\\mathcal H(t)\\Psi$ with a time-dependent Hamiltonian $\\mathcal H(t)$. Using this, we show that when we take $\\mathcal H(t)=-\\Delta+q(t,x)$ restricted to a finite box within $\\mathbb Z^d$, where $q(t,x)$ is a possibly time-dependent Gaussian process, $\\mathbf E\\big[|\\Psi(t,x)|^2\\big]$ can be written as a relatively simple expectation that involves self- and mutual-intersections of random walks. In particular, this formula hints at a unified combinatorial mechanism that explains the occurrence of localization for both time-dependent and time-independent noises.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-14T21:55:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vector-valued schrödinger semigroups",
      "probabilistic representation",
      "ordered exponentials",
      "anderson localization",
      "feynman-kac formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}