Filters

2015 - 20185

More filters

20195
Reset filters

Settings

Author: Santos, Renato Soares dos ×

Search Results

Now showing 1 - 5 of 5
  • Thumbnail Image
    Item
    Brownian motion in attenuated or renormalized inverse-square Poisson potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Nelson, Peter; Santos, Renato Soares dos
    We consider the parabolic Anderson problem with random potentials having inverse-square singularities around the points of a standard Poisson point process in ℝ d, d ≥3. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel 𝔎 behaving as 𝔎 (x)≈ Θ x -2 near the origin, where Θ ∈(0,(d-2)2/16]. In order to make sense of the corresponding path integrals, we require the potential to be either attenuated (meaning that 𝔎 is integrable at infinity) or, when d=3, renormalized, as introduced by Chen and Kulik in [8]. Our main results include existence and large-time asymptotics of non-negative solutions via Feynman-Kac representation. In particular, we settle for the renormalized potential in d=3 the problem with critical parameter Θ = 1/16, left open by Chen and Rosinski in [9].
  • Thumbnail Image
    Item
    Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Biskup, Marek; König, Wolfgang; Santos, Renato Soares dos
    We study the solutions to the Cauchy problem on the with random potential and localised initial data. Here we consider the random Schrödinger 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.
  • Thumbnail Image
    Item
    The Bouchaud-Anderson model with double-exponential potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Muirhead, Stephen; Pymar, Richard; Santos, Renato Soares dos
    The Bouchaud-Anderson model (BAM) is a generalisation of the parabolic Anderson model (PAM) in which the driving simple random walk is replaced by a random walk in an inhomogeneous trapping landscape; the BAM reduces to the PAM in the case of constant traps. In this paper we study the BAM with double-exponential potential. We prove the complete localisation of the model whenever the distribution of the traps is unbounded. This may be contrasted with the case of constant traps (i.e. the PAM), for which it is known that complete localisation fails. This shows that the presence of an inhomogeneous trapping landscape may cause a system of branching particles to exhibit qualitatively distinct concentration behaviour.
  • Thumbnail Image
    Item
    Zero-one law for directional transience of one-dimensional random walks in dynamic random environments
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Orenshtein, Tal; Santos, Renato Soares dos
    We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d. in space and Markovian in time.
  • Thumbnail Image
    Item
    Random walk on random walks: Higher dimensions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Blondel, Oriane; Hilário, Marcelo R.; Santos, Renato Soares dos; Sidoravicius, Vladas; Teixeira, Augusto
    We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].