Searcharxiv⌕ Search

arXiv subjects

Antoine Mouzard

Publications and source records attributed to Antoine Mouzard.

12 recordsLinked to original sources
arXiv2026

Nonlinear Schrödinger equations with spatial white noise potential on full space for $d\le 3$

In this paper, we prove existence and uniqueness of energy solutions for nonlinear Schrödinger equations with a multiplicative white noise on $R^d$ with $d\le3$. We rely on an exponential trans-form and conserved quantities for existence of energy solutions. Using paracontrolled calculus, we prove Strichartz inequalities which encode the dispersive properties of the solutions. This allows to obtain local well-posedness for low regularity solutions and uniqueness of energy solutions for various equations. In particular, our results are the first results of propagation without loss of both regularity and localization for such equations in full space as well as the first results on $R^3$ for such singular dispersive SPDEs. We are also obtain local well-posedness in two dimensions for deterministic initial data.

math.PR↗
arXiv2025

High order uniform in time schemes for weakly nonlinear Schrödinger equation and wave turbulence

We introduce two multiscale numerical schemes for the time integration of weakly nonlinear Schrödinger equations, built upon the discretization of Picard iterates of the solution. These high-order schemes are designed to achieve high precision with respect to the small nonlinearity parameter under particular CFL condition. By exploiting the scattering properties of these schemes thanks to a low-frequency projected linear flow, we also establish its uniform accuracy over long time horizons. Numerical simulations are provided to illustrate the theoretical results, and these schemes are further applied to investigate dynamics in the framework of wave turbulence.

math.NA↗
arXiv2025

Ergodicity of the Anderson $Φ_2^4$ model

We consider the parabolic stochastic quantization equation associated to the $Φ_2^4$ model on the torus in a spatial white noise environment. We study the long time behavior of this heat equation with independent multiplicative white noise and additive spacetime white noise, which is a singular SPDE in a singular environement and requires two different renormalization procedures. We prove that the solution is global in time with a strong a priori $L^p$ bound independent of the initial data in $C^{-\varepsilon}$ for large $p$. The quenched solution given the environment is shown to be an infinite dimensional Markov process which satisfies the strong Feller property. We prove exponential convergence to a unique invariant measure using a Doeblin criterion for the transition semigroup. In particular, our work is a generalization of a previous work by Tsatsoulis and Weber in a case which is not translation invariant hence the method makes no use of the reversibility of the dynamics or the explicit knowledge of the invariant measure and it is therefore in principle applicable to situations where these are not available such as the vector-valued case.

math.PR↗
arXiv2024

Periodic nonlinear Schrödinger equation with distributional potential and invariant measures

In this paper, we continue some investigations on the periodic NLSE started by Lebowitz, Rose and Speer and by Bourgain with the addition of a distributional multiplicative potential. We prove that the equation is globally wellposed for a set of data of full normalized Gibbs measure, after suitable truncation in the focusing case. The set and the measure are invariant under the flow. The main ingredients used are Strichartz estimates on periodic NLS with distributional potential to obtain local well-posedness for low regularity initial data.

math.AP↗
arXiv2024

Anderson stochastic quantization equation

We study the parabolic defocusing stochastic quantization equation with both mutliplicative spatial white noise and an independant space-time white noise forcing, on compact surfaces, with polynomial nonlinearity. After renormalizing the nonlinearity, we construct the random Gibbs measure as an absolutely continuous measure with respect to the law of the Anderson Gaussian Free Field for fixed realization of the spatial white noise. Then, when the initial data is distributed according to the Gibbs measure, we prove almost sure global well-posedness for the dynamics and invariance of the Gibbs measure.

math.AP↗
arXiv2023

A simple construction of the Anderson operator via its quadratic form in dimensions two and three

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 regularity structures or paracontrolled calculus which are usually used for the construction of the operator. The knowledge of the form is robust enough to deduce important properties such as positivity and irreducibility of the corresponding semigroup. The latter property gives existence of a spectral gap.

math.AP↗
arXiv2023

The logarithmic Schr{ö}dinger equation with spatial white noise on the full space

We solve the Schr{ö}dinger equation with logarithmic nonlinearity and multiplicative spatial white noise on R d with d $\le$ 2. Because of the nonlinearity, the regularity structures and the paracontrolled calculus can not be used. To solve the equation, we rely on an exponential transform that has proven useful in the context of other singular SPDEs.

math.AP↗
arXiv2023

Scattering, random phase and wave turbulence

We start from the remark that in wave turbulence theory, exemplified by the cubic twodimensional Schr{ö}dinger equation (NLS) on the real plane, the regularity of the resonant manifold is linked with dispersive properties of the equation and thus with scattering phenomena. In contrast with classical analysis starting with a dynamics on a large periodic box, we propose to study NLS set on the real plane using the dispersive effects, by considering the time evolution operator in various time scales for deterministic and random initial data. By considering periodic functions embedded in the whole space by gaussian truncation, this allows explicit calculations and we identify two different regimes where the operators converges towards the kinetic operator but with different form of convergence.

math.AP↗
arXiv2022

The infinitesimal generator of the Brox diffusion

We construct the infinitesimal generator of the Brox diffusion on a line with a periodic Brownian environment. This gives a new construction of the process and allows to solve the singular martingale problem. We prove that the associated semigroup is strong Feller with Gaussian lower and upper bounds. This also yields a construction of the Brox diffusion on a segment with periodic or Dirichlet boundary conditions. In this bounded space, we prove that there exists a unique measure and the existence of a spectral gap giving exponential ergodicity of the diffusion.

math.PR↗
arXiv2022

Strichartz inequalities with white noise potential on compact surfaces

We prove Strichatz inequalities for the Schr{ö}dinger equation and the wave equation with multiplicative noise on a two-dimensional manifold. This relies on the Anderson Hamiltonian H described using high order paracontrolled calculus. As an application, it gives a low regularity solution theory for the associated nonlinear equations.

math.AP↗
arXiv2021

2D random magnetic Laplacian with white noise magnetic field

We define the random magnetic Laplacien with spatial white noise as magnetic field on the two-dimensional torus using paracontrolled calculus. It yields a random self-adjoint operator with pure point spectrum and domain a random subspace of nonsmooth functions in L 2. We give sharp bounds on the eigenvalues which imply an almost sure Weyl-type law.

math-ph↗