Séminaire d'analyse (archives)

Consider a particle randomly moving in a bounded (planar) domain starting at any given point within. Assume it bounces against the boundary and consider $\Sigma$, a small part of that boundary. What is the expected time we need to wait before the particle hits $\Sigma$ ? This question is known as the narrow escape problem. We can also consider the related question : what is the probability that the particle hits $\Sigma$ before another given subset of the boundary $\Gamma$ ? In this talk, I will address these questions and give quantitative answers in the asymptotic regime where the lengths of the windows tend to 0. To tackle the problem, I will prove a Feynman-Kac formula, linking the stochastic process studied to a deterministic PDE which has the form of a Poisson equation with mixed boundary conditions. Then, constructing appropriate quasimodes to this PDE, we are able to derive sharp asymptotics for the expected time and probabilities.

The full regularity of harmonic maps from a given surface into an arbitrary Riemannian manifold has been proved by Hélein in 1991. This is not true anymore when the domain has dimension strictly greater than 2, Rivière constructed an example of harmonic map from a 3-dimensional domain which is everywhere discontinuous in 1995. There are many possible generalizations of these maps to the higher dimensional case in order to recover the regularity of some "optimal" maps. For most of these generalizations, the optimal regularity in full generality is still open. In this talk, we will discuss some recent progress obtained for n-harmonic maps. This is a joint work with Armin Schikorra.

Quantum mechanics is well approximated by classical physics when Planck's constant is considered small, i.e., in the semi-classical limit. Typically, one can study an observable associated with a particle, such as its momentum or its position, and show that its dynamics is given by classical dynamics at first order, with corrections of the order of Planck's constant. In this talk, I will present more precisely the concept of semi-classical limits, the standard mathematical results known for non-relativistic quantum mechanics, and my work that concerns the semi-classical limit in the context of relativistic quantum mechanics. Concretely, I will show how to adapt the modulated energy method to the Klein-Gordon and Klein-Gordon-Maxwell equations and how to recover relativistic mechanics (instead of classical mechanics) at the semi-classical limit

We consider a system of N Brownian particles interacting through a long-range smooth potential. It is known that "propagation of chaos" holds in the mean-field scaling. Assume indeed that the initial distribution of the particles is chaotic, i.e. that the particles are independent and identically distributed. Then, for any given time, and as N becomes large, the distribution of particles remains chaotic. Moreover, the distribution of a typical particle is given by the solution of a Vlasov-Fokker-Planck equation.

In this talk, we will investigate the creation of chaos phenomenon. Starting from an initial distribution of particles which is only exchangeable, we prove that in some weak norm, propagation of chaos holds up to an error stemming from initial correlations, exponentially damped over time. This is a joint work with Armand Bernou and Mitia Duerinckx.

We consider a tagged particle in mean field interaction with a Rayleigh gas of density N, and prove the convergence of its trajectory, as N goes to infinity, to the one of a diffusion process associated with the linear Landau equation. This is joint work with T. Bodineau.

On s'intéresse à un système de N particules dans le régime de champ moyen, c'est-à-dire avec des interactions de faible intensité (d'ordre 1/N) mais à longue distance (d'ordre 1). Sur les temps courts (d'ordre 1), ce système est décrit par l'équation de Vlasov, qui est conservative. Sur des temps beaucoup plus longs (d'ordre N), par contre, la théorie de Lenard-Balescu prédit la relaxation du système vers l'équilibre (au sens fort, avec de la dissipation d'entropie). Formellement, cette relaxation lente s'explique par les corrélations entre les particules. Cependant, aucune justification rigoureuse n'a été obtenue à ce jour: les seuls résultats sont des preuves de consistance (dérivation au temps 0).

Dans cet exposé, nous revisitons le problème en partant d'un modèle simplifié, qui s'inspire de la version linéaire du problème (due à Duerinckx et Saint-Raymond) et dans lequel la hiérarchie BBGKY est tronquée à un ordre arbitraire. Partant d'une approche perturbative en diagrammes de Feynman, utilisant des idées de renormalisation, et des estimations hypoelliptiques pour les propagateurs renormalisés, nous parvenons à atteindre le temps cinétique (d'ordre N) pour ce modèle.

In this talk, we discuss the effects of perturbations on the topology and geometry of nodal sets/zero sets of Laplace eigenfunctions. A conjecture by Payne states that the nodal set of the second Dirichlet eigenfunction on a bounded planar domain intersects the boundary at exactly two points. We will look into certain stability properties of the nodal sets and discuss some recent results concerning the conjecture. Then, utilising the stability properties, we will observe the prescription of nodal data on Riemannian surfaces, focusing on the following two aspects: the construction of eigenfunctions with a prescribed number of nodal intersections at the boundary, and the realisation of Courant-sharp eigenfunctions at arbitrarily high levels.

Dans cet exposé, je présenterai deux modèles jouets permettant de traiter deux difficultés dans l'étude de l'équation de Vlasov (pour les photons) sur un trou noir, liées notamment à l'existence d'orbites piégées. Le fait que l'ensemble de ces orbites soit normalement hyperboliques permet de contrôler certaines dérivées radiales des solutions ainsi que de montrer que le flux d'énergie décroit exponentiellement.

In this overview talk, I will present the encounter-based approach to diffusive processes in Euclidean domains and highlight its fundamental relation to the Steklov spectral problem. So, the Steklov eigenfunctions turn out to be particularly useful for representing heat kernels with Robin boundary condition and describing diffusive dynamics with reaction events on the boundary. In the second part of the talk, I will discuss the asymptotic behavior of the Steklov eigenvalues for the exterior Steklov problem. Some open questions related to spectral, probabilistic and asymptotic aspects of this problem will be outlined.

References:

[1] D. S. Grebenkov, Paradigm Shift in Diffusion-Mediated Surface Phenomena, Phys. Rev. Lett. 125, 078102 (2020).

[2] D. S. Grebenkov and A. Chaigneau, The Steklov problem for exterior domains: asymptotic behavior and applications (accepted to J. Math. Phys.; preprint ArXiv 2407.09864v2)