WIAS Preprint No. 2295, (2016)

Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails


Authors

  • Biskup, Marek
  • König, Wolfgang
    ORCID: 0000-0002-7673-4364
  • Soares dos Santos, Renato

2010 Mathematics Subject Classification

  • 60J55 60F10 60K35 60K37 82B44

Keywords

  • heat equation with random coefficients, random Schrödinger operator, Feynman-Kac formula, Anderson localisation, mass concentration, spectral expansion, eigenvalue order statistics

DOI

10.20347/WIAS.PREPRINT.2295

Abstract

We study the solutions to the Cauchy problem on the with random potential and localised initial data. Here we consider the random Schr?dinger operator, i.e., the Laplace operator with random field, whose upper tails are doubly exponentially distributed in our case. We prove that, for large times and with large probability, a majority of the total mass of the solution resides in a bounded neighborhood of a site that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian and the distance to the origin. The processes of mass concentration and the rescaled total mass are shown to converge in distribution under suitable scaling of space and time. Aging results are also established. The proof uses the characterization of eigenvalue order statistics for the random Schrödinger operator in large sets recently proved by the first two authors.

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