Filters

2010 - 20204

More filters

2019 - 201922020 - 20222
Reset filters

Settings

Author: König, Wolfgang × Subject: Parabolic Anderson model ×

Search Results

Now showing 1 - 4 of 4
  • Thumbnail Image
    Item
    The parabolic Anderson model on a Galton--Watson tree
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) den Hollander, Frank; König, Wolfgang; Soares dos Santos, Renato
    We study the long-time asymptotics of the total mass of the solution to the parabolic Anderson model ( PAM) on a supercritical Galton-Watson random tree with bounded degrees. We identify the second-order contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a tree with minimal degree. Our approach can be applied to finite locally tree-like random graphs, in a coupled limit where both time and graph size tend to infinity. As an example, we consider the configuration model or, more precisely, the uniform simple random graph with a prescribed degree sequence.
  • Thumbnail Image
    Item
    Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Biskup, Marek; König, Wolfgang
    We consider random Schrödinger operators of the form Delta+zeta , where D is the lattice Laplacian on Zd and Delta is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of Zd. We show that for sigma with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class. The corresponding eigenfunctions are exponentially localized in regions where zeta takes large, and properly arranged, values. A new and self-contained argument is thus provided for Anderson localization at the spectral edge which permits a rather explicit description of the shape of the potential and the eigenfunctions. Our study serves as an input into the analysis of an associated parabolic Anderson problem.
  • Thumbnail Image
    Item
    The parabolic Anderson model with acceleration and deceleration
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) König, Wolfgang; Schmidt, Sylvia
    We describe the large-time moment asymptotics for the parabolic Anderson model where the speed of the diffusion is coupled with time, inducing an acceleration or deceleration. We find a lower critical scale, below which the mass flow gets stuck. On this scale, a new interesting variational problem arises in the description of the asymptotics. Furthermore, we find an upper critical scale above which the potential enters the asymptotics only via some average, but not via its extreme values. We make out altogether five phases, three of which can be described by results that are qualitatively similar to those from the constant-speed parabolic Anderson model in earlier work by various authors. Our proofs consist of adaptations and refinements of their methods, as well as a variational convergence method borrowed from finite elements theory.
  • Thumbnail Image
    Item
    Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) König, Wolfgang; Perkowski, Nicolas; van Zuijlen, Willem
    We consider the parabolic Anderson model (PAM) in ℝ ² with a Gaussian (space) white-noise potential. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time t is given asymptotically by Χ t log t, with the deterministic constant Χ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour principal Dirichlet of the eigenvalue the Anderson operator on the t by t box around zero asymptotically by Χ log t.