Abstract We define the Anderson Hamiltonian H on a two-dimensional manifold using the
high order paracontrolled calculus. It is a self-adjoint operator with pure point spectrum. We …

arXiv:2208.09352v1 [math.AP] 19 Aug 2022 Page 1 arXiv:2208.09352v1 [math.AP] 19 Aug 2022
Anderson Hamiltonian and associated Nonlinear Stochastic Wave and Schrödinger equations …

We consider the continuous Anderson operator $ H=\Delta+\xi $ on a two dimensional
closed Riemannian manifold $\mathcal {S} $. We provide a short self-contained functional …

We consider the parabolic Anderson model (PAM)∂ tu= 1 2 Δ u+ ξ u in R 2 with a Gaussian
(space) white-noise potential ξ. We prove that the almost-sure large-time asymptotic …

We consider the nonlinear Schrödinger equation with multiplicative spatial white noise and
an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove …

Using the method of paracontrolled distributions, we show the local well-posedness of an
additive noise approximation to the fluctuating hydrodynamics of the Keller-Segel model on …

We develop further in this work the high order paracontrolled calculus setting to deal with the
analytic part of the study of quasilinear singular PDEs. Continuity results for a number of …

We consider the random Schrödinger operator on R obtained by perturbing the Laplacian
with a white noise. We prove that Anderson localization holds for this operator: almost surely …

We provide a simple construction of the Anderson operator in dimensions two and three.
This is done through its quadratic form. We rely on an exponential transform instead of the …

We study the Gaussian measure whose covariance is related to the Anderson Hamiltonian
operator, proving that it admits a regular coupling to the (standard) Gaussian free field …