Browsing by Author "van Zuijlen, Willem"

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    Asymptotics of the eigenvalues of the Anderson Hamiltonian with white noise potential in two dimensions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Chouk, Khalil; van Zuijlen, Willem
    In this paper we consider the Anderson Hamiltonian with white noise potential on the box [0,L]² with Dirichlet boundary conditions. We show that all the eigenvalues divided by log L converge as L → ∞ almost surely to the same deterministic constant, which is given by a variational formula.
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    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.
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    Quantitative heat kernel estimates for diffusions with distributional drift
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Perkowski, Nicolas; van Zuijlen, Willem
    We consider the stochastic differential equation on ℝ d given by d X t = b(t,Xt ) d t + d Bt, where B is a Brownian motion and b is considered to be a distribution of regularity > - 1/2. We show that the martingale solution of the SDE has a transition kernel Γt and prove upper and lower heat kernel bounds for Γt with explicit dependence on t and the norm of b.
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    Quantitative Heat-Kernel Estimates for Diffusions with Distributional Drift
    (Dordrecht [u.a.] : Springer Science + Business Media B.V, 2022) Perkowski, Nicolas; van Zuijlen, Willem
    [For Abstract, see PDF]