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Anne-Sophie de Suzzoni

Publications and source records attributed to Anne-Sophie de Suzzoni.

At least 19 recordsLinked to original sources
arXiv2026

Obstruction to quasi-invariance of Gaussian measures under transport flows on Riemannian Manifolds

We prove an obstruction to quasi-invariance of Gaussian fields under divergence free transport flows on Riemannian manifolds. For the centered Gaussian field with covariance $(1-Δ_g)^{-α}$, $α>\frac{d}{2}$, quasi-invariance under the transport flow is equivalent to invariance. This happens precisely when the flow acts by isometries, or equivalently when the underlying vector field is Killing. The proof leverages the Feldman--Hájek theorem and utilizes localized pseudo-differential computations. On $\mathbb R^d$ this leaves only rigid motions, while on flat tori this leaves only translations.

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arXiv2026

Construction of a Gibbs measure for the zonal Dirac equation

We propose a framework to construct Gibbs measures for the Dirac equation. We consider the Dirac equation on the sphere with a "Hartree-type" nonlinearity. We consider a zonal model, that is the analog of a spherically symmetric model but on the sphere. We build a Gibbs measure for this model. With a compactness argument, we prove the existence of a random variable that is a weak solution to the Dirac equation and whose law is the Gibbs measure at all times.

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arXiv2025

Wave turbulence for a semilinear Klein-Gordon system

In this article we consider a system of two Klein-Gordon equations, set on the $d$-dimensional box of size $L$, coupled through quadratic semilinear terms of strength $\varepsilon$ and evolving from well-prepared random initial data. We rigorously derive the effective dynamics for the correlations associated to the solution, in the limit where $L\to\infty$ and $\varepsilon\to 0$ according to some power law. The main novelty of our work is that, due to the absence of invariances, trivial resonances always take precedence over quasi-resonances. The derivation of the nonlinear effective dynamics is justified up time to $δT$ , where $T =\varepsilon^{-2}$ is the appropriate timescale and $δ$ is independent of $L$ and $\varepsilon$. We use Feynmann interaction diagrams, here adapted to a normal form reduction and to the coupled nature of our real-valued system. We also introduce a frequency decomposition at the level of the diagrammatic and develop a new combinatorial tool which allows us to work with the Klein-Gordon dispersion relation.

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arXiv2024

Stability of homogeneous equilibria of the Hartree-Fock equation, for its equivalent formulation for random fields

The Hartree-Fock equation admits homogeneous states that model infinitely many particles at equilibrium. We prove their asymptotic stability in large dimensions, under assumptions on the linearised operator. Perturbations are moreover showed to scatter to linear waves. We obtain this result for the equivalent formulation of the Hartree-Fock equation in the framework of random fields. The main novelty is to study the full Hartree-Fock equation, including for the first time the exchange term in the study of these stationary solutions.

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arXiv2022

Strichartz estimates for the Dirac equation on asymptotically flat manifolds

In this paper we prove Strichartz estimates for the Dirac equation on asymptotically flat manifolds. The proof combines the weak dispersive estimates proved by the first two authors with the Strichartz and smoothing estimates for the wave and Klein-Gordon flows, exploiting some recent results in the same geometrical setting.

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arXiv2021

Large time well posedness for a Dirac--Klein-Gordon system

In this paper we prove well posedness for a system coupling a nonlinear Dirac with a Klein-Gordon equation that represents a toy model for the Helium atom with relativistic corrections: the wave function of the electrons interacts with an electric field generated by a nucleus with a given charge density. One of the main ingredients we need is a new family of Strichartz estimates for time dependent perturbations of the Dirac equation: these represent a result of independent interest.

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arXiv2021

Global Strichartz estimates for the Dirac equation on symmetric spaces

In this paper we study global-in-time, weighted Strichartz estimates for the Dirac equation on warped product spaces in dimension $n\geq3$. In particular, we prove estimates for the dynamics restricted to eigenspaces of the Dirac operator on the compact spin manifolds defining the ambient manifold under some explicit sufficient condition on the metric, and estimates with loss of angular derivatives for general initial data in the setting of spherically symmetric and asymptotically flat manifolds.

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arXiv2020

Singularities in the weak turbulence regime for the quintic Schrödinger equation

In this paper, we discuss the problem of derivation of kinetic equations from the theory of weak turbulence for the quintic Schrödinger equation. We study the quintic Schrödinger equation on $L\mathbb T$, with $L\gg 1$ and with a non-linearity of size $\varepsilon\ll 1$. We consider the correlations $f(T)$ of the Fourier coefficients of the solution at times $t = T\varepsilon^{-2}$ when $\varepsilon\rightarrow 0$ and $L\rightarrow \infty$. Our results can be summed up in the following way : there exists a regime for $\varepsilon$ and $L$ such that for $T$ dyadic, $f(T)$ has the form expected from the physics literature, but such that $f$ has an infinite number of discontinuity points.

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arXiv2020

Strichartz estimates for the Klein-Gordon equation in a conical singular space

Consider a conical singular space $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. We study the Klein-Gordon equations with inverse-square potentials in the space $X$, proving in particular global-in-time Strichartz estimates in this setting.

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arXiv2020

Stability of Steady States for Hartree and Schrodinger Equations for Infinitely Many Particles

We prove a scattering result near certain steady states for a Hartree equation for a random field. This equation describes the evolution of a system of infinitely many particles. It is an analogous formulation of the usual Hartree equation for density matrices. We treat dimensions 2 and 3, extending our previous result. We reach a large class of interaction potentials, which includes the nonlinear Schrodinger equation. This result has an incidence in the density matrices framework. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrodinger equation, and on the use of explicit low frequency cancellations as in the work of Lewin and Sabin. To relate to density matrices, we use Strichartz estimates for orthonormal systems from Frank and Sabin, and Leibniz rules for integral operators.

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arXiv2019

Invariance of Gibbs measures under the flows of Hamiltonian equations on the real line

We prove that the Gibbs measures $ρ$ for a class of Hamiltonian equations written $\partial_t u = J (-\triangle u + V'(|u|^2)u)$ on the real line are invariant under the flow of this equation in the sense that there exist random variables $X(t)$ whose laws are $ρ$ (thus independent from $t$) and such that $t\mapsto X(t)$ is a solution to the above equation. Besides, for all $t$, $X(t)$ is almost surely not in $L^2$ which provides as a direct consequence the existence of weak solutions for initial data not in $L^2$. The proof uses Prokhorov's theorem, Skorohod's theorem, as in the strategy in \cite{burqtzv} and Feynman-Kac's integrals.

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arXiv2019

A Dirac field interacting with point nuclear dynamics

The system describing a single Dirac electron field coupled with classically moving point nuclei is presented and studied. The model is a semi-relativistic extension of corresponding time-dependent one-body Hartree-Fock equation coupled with classical nuclear dynamics, already known and studied both in quantum chemistry and in rigorous mathematical literature. We prove local existence of solutions for data in $H^s$ with $s>1$ and local well posedness in $H^s$ for $s>3/2$. In the course of the analysis a second new result of independent interest is discussed and proved, namely the construction of the propagator for the Dirac operator with several moving Coulomb singularities.

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arXiv2018

Stability of equilibria for a Hartree equation for random fields

We consider a Hartree equation for a random variable, which describes the temporal evolution of infinitely many Fermions. On the Euclidean space, this equation possesses equilibria which are not localised. We show their stability through a scattering result, with respect to localised perturbations in the defocusing case in high dimensions $d\geq 4$. This provides an analogue of the results of Lewin and Sabin \cite{LS2}, and of Chen, Hong and Pavlović \cite{CHP2} for the Hartree equation on operators. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrödinger and Gross-Pitaevskii equations.

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arXiv2016

On Gibbs measure and weak flow for the cubic NLS with non-localised initial data

In this paper we prove the existence of an invariant measure for the cubic NLS $$i\partial_t u + \bigtriangleup u - |u|^2 u = 0$$ on the real line in the sense that we prove the existence of a measure $ρ$ supported by non-localised functions such that there exists random variables $X(t)$ whose laws are $ρ$ (thus independent of $t$) and such that $t\mapsto X(t)$ is a solution to the cubic NLS. Our strategy for the proof is inspired by \cite{burqtzv} and relies on the application of Prokhorov and Skorokhod Theorems to a sequence of measures which are invariant under some approximating flows, as we proved in our previous \cite{lastbaby}. However, the work by Bourgain, \cite{B00} provides a stronger result than this one, as it gives almost sure strong solutions for the cubic NLS and the invariance of the measure can be deduced from it.

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