EDP et Probabilité

26 oct. 2022, 11:00 → 28 oct. 2022, 17:00 Europe/Paris

Institut de Mathématiques de Bordeaux

Conférenciers:

Marc Arnaudon   (Université de Bordeaux)

Beniamin Bogosel   (Ecole Polytechnique)

Stéphane Brull   (Université de Bordeaux)

Laurent Bruneau   (Université de Cergy-Pontoise)

Gaëtan Cane   (Université d'Orléans)

Persi Diaconis*   (Stanford University)

Pierre Gervais   (ENS Paris)

Antoine Gloria*  (Sorbonne Université)

Bernard Helffer   (Université de Nantes)

Antoine Mouzard   (Université de Rennes)

Adrien Richou   (Université de Bordeaux)

Constanza Rojas-Molina   (Université de Cergy-Pontoise)

Martin Slowik   (Université de Mannheim)

Martin Vogel   (Université de Strasbourg)

 

* = par zoom

 

 

 

mercredi 26 octobre

12:00 Buffet + Accueil

1 C. Rojas-Molina: Open questions for fractional random Schrödinger operators

We review some recent results on the fractional Anderson model, a random Schrödinger operator driven by a fractional laplacian. The interest on the latter lies in their association to stable Levy processes, random walks with long jumps and anomalous diffusion. While in certain regimes, the standard proofs of localization break down in this setting, we can still gather information about the integrated density of states and obtain estimates on the decay of the Green’s function through a link to long-range self-avoiding random walks that exists in the case of random perturbations of the Laplacian.

This is based on joint work with M. Gebert, and with M. Disertori and R. Maturana Escobar.

2 G. Cane: Superdiffusion transition for a noisy harmonic chain subject to a magnetic field

Understanding the diffusive or superdiffusive behavior of the energy in classical physical systems is challenging because of the non-linearity of the interactions between the particles. A way to reduce the difficulty is to replace the nonlinearity by a stochastic noise. In this presentation I will consider a noisy harmonic chain subjected to a magnetic field. We will see that according to the intensity of the magnetic field, the superdiffusive nature of the system changes.

16:00 Pause café

3 A. Richou: About conditional expectation in a non convex set

In this talk, I will explain how to define a good notion of conditional expectation constrained to live in a non convex smooth domain of $R^d$. I will present some existence and uniqueness results obtained with J.-F. Chassagneux (Univ. Paris Cité) and S. Nadtochiy (IIT Chicago) under appropriate assumptions. I will also explain how to improve our results in dimension 2 by using a more geometric approach. This last part is a work in progress with M. Arnaudon (Univ. Bordeaux), J.-F. Chassagneux (Univ. Paris Cité) and S. Nadtochiy (IIT Chicago).

jeudi 27 octobre

4 A. Gloria - ZOOM - The landscape function in Rd

In this talk I will describe the landscape function, a quantity introduced by Filoche and Mayboroda ten years ago in order to study localization phenomena for random Schrödinger operators. In the first part I will quickly discuss the striking fact that the landscape function ac curately predicts the support of localized eigenstates and their energy in bounded domains. In the main part of the talk I will show how to define the landscape function in the whole space, and characterise its correlations. The main ingredient is the exponential decay of the Green function of the random Schrödinger oper-
ator, which I shall establish.

This is a joint work with Guy David and Svitlana Mayboroda.

10:00 Pause café

5 S. Brull: Fredholm property of the linearized Boltzmann operator for a mixture of polyatomic gases

In this talk I will present the proof of the Fredholm alternative for the linearized Boltzmann operator. The model is describded with a distribution function with an additional continous energy variable. The collision operator is based on the Borgnakke-Larsen procedure. We present the proof in the case of a single gas and of mixtures. The cornerstone of the proof is the introduction of a kernel on the perturbation part in order to prove that the operator is Hilbert Schmidt.

6 P. Gervais: On the Boltzmann equation for long-range interactions close to equilibrium

The Boltzmann equation, introduced by J.C. Maxwell and L. Boltzmann at the end of the 19th century, describes the evolution of a gas at the molecular level using a statistical point of view. More precisely, instead of considering the exact position and velocity of each of the particles making up the gas, we are interested in their statistical distributions for a typical particle. One of the main mathematical difficulties of this equation comes from the interactions between pairs of ”distant” particles. Considered individually, they have little influence on the velocities of the particles, but are extremely frequent, which results in the presence of an ”angular singularity” in the operator modeling their effect. This difficulty is responsible for the very slow evolution of the mathematical theory of the Boltzmann equation. In 1963, H. Grad proposed a way to neglect this singularity, leading to a rapid progress in our understanding of this equation. This angular singularity is however not insignificant, it provides among other things a regularizing effect to the equation, and has been studied in many works since the 1990s. In this talk I will present how to construct solutions to the Boltzmann equation close to equilibrium.

12:30 Déjeuner

7 M. Arnaudon: Coupling of Brownian motions with set valued dual processes on Riemannian manifolds

In this talk we will motivate and explain the evolution by renormalized stochastic mean curvature flow, of boundaries of relatively compact connected domains in a Riemannian manifolds. We will construct coupled Brownian motions inside the moving domains, satisfying a Markov intertwining relation. We will prove that the Brownian motions perform perfect simulation of uniform law, when the domain reaches the whole manifold. We will investigate the example of evolution of discs in spheres, and of symmetric domains in the Euclidean plane. Skeletons of moving domains will play a major role.

8 A. Mouzard: A random continuum polymer associated to the Anderson Hamiltonian on compact surfaces

In this talk, I will present the construction of a random continuum polymer in the presence of a spatial white noise on compact surfaces. This relies on the continuous Anderson Hamiltonian, that is the random Schrödinger operator with white noise potential. The description of the spectral properties of this random operator is obtained using the tools of paracontrolled calculs recently developed to solve singular stochastic PDEs.

16:30 Pause café

9 P. Diaconis - ZOOM - Random walk on the Rado graph and Hardy’s inequality for trees

The Rado graph $R$ is a natural limit of the set of all finite graphs. One way to think of it is: on ${\mathbb N}$ (natural numbers) flip a fair coin for each pair of vertices and put an edge in if it comes up heads. Each vertex has infinite degree and the diameter is $2$. In joint work with Sourav Chatterjee and Laurent Miclo we study a natural laplacian: pick a positive probability $Q(j)$ on ${\mathbb N}$. From $i$, the walk chooses a nearest neighbor of $i$ (in $R$) with probability proportional to $Q(j)$.This walk has a stationary distribution and one may ask about rates of convergence to stationarity. The main result studies $Q(i) = 1/2^{(i+1)}$ and shows that, starting from $i$, $ {\rm log}^*_2 i$ steps suffice for convergence and, are needed for infinitely many $i$. The analysis uses a novel form of weighted Hardy inequalities for trees; Hardy’s inequalities with weights are familiar on ${\mathbb R}$ but even on ${\mathbb R}^{d}$ are a poorly developed tool. We develop the version on infinite trees and use it to get a spectral gap for the walk and to give a new picture of the geometry of the graph $R$. I will try to explain $R$ and Hardy’s inequalities (and the application) in ’mathematical English’. Understanding the problem for other $Q$ (eg $Q(j) = 1/j^2 $) is open.

20:00 Dîner au "Le Plana" - 22 Place de la Victoire - Bordeaux

vendredi 28 octobre

10 B. Helffer: Quantum tunneling in deep potential wells and strong magnetic field revisited

Inspired by a recent paper by C. Fefferman, J. Shapiro and M. Weinstein, we investigate quantum tunneling for a Hamiltonian with a symmetric double well and a uniform magnetic field. In the simultaneous limit of strong magnetic field and deep potential wells with disjoint supports, tunneling occurs and we derive accurate estimates of its magnitude.

This talk is based on a joint work with A. Kachmar.

10:00 Pause café

11 B. Bogosel: On the polygonal Faber-Krahn inequality

It has been conjectured by Pólya and Szegö in 1951 that among n-gons with fixed area the regular one minimizes the first eigenvalue of the Dirichlet-Laplace operator. Despite its apparent simplicity, this result has only been proved for triangles and quadrilaterals. In this work we show that the proof of the conjecture can be reduced to finitely many certified numerical computations. Moreover, the local minimality of the regular polygon is reduced to a single validated numerical computation.

The steps of the proof strategy include the analytic computation of the Hessian matrix of the first eigenvalue, the stability of the Hessian with respect to vertex perturbations and analytic upper bounds for the diameter of an optimal set. Explicit a priori error estimates are given for the finite element computation of the eigenvalues of the Hessian matrix of the first eigenvalue associated to the regular polygon.

Results presented are obtained in collaboration with Dorin Bucur.

12 M. Vogel: Eigenvector localization and delocalization of noisy non selfadjoint operators

It is now very well established that small random perturbations lead to probabilistic Weyl laws for the eigenvalue asymptotics of non-selfadjoint semiclassical pseudo-differential operators, Berezin-Toeplitz quantizations of compact Kähler manifolds and Toeplitz matrices. In this talk, I will I present recent work in collaboration with Anirban Basak and Ofer Zeitouni on eigenvector localization and delocalization in the model case of large non-selfadjoint Toeplitz matrices with small random perturbations.

12:30 Déjeuner au "Yamato" de Talence

13 M. Slowik: Metastability of Glauber dynamics with inhomogeneous coupling disorder

Metastability is a phenomenon that occurs in the dynamics of a multi-stable non-linear system subject to noise. It is characterised by the existence of multiple, well separated time scales. The talk will be focused on the metastable behaviour of a general class of mean-field-like spin systems with random couplings that evolve according to a Glauber dynamics at fixed temperature. This class of systems comprises both the Ising model on inhomogeneous dense random graphs and the randomly diluted Hopfield model. Assuming that the corresponding system in which the random couplings are replaced by their averages is metastable I will explain how the metastability of the random system is implied with high probability. In particular, I will discuss the tail behaviour of the relevant metastable hitting times of the two systems and the moments of their ratio.

This is joint work with A. Bovier, F. Den Hollander, S. Marello and E. Pulvirenti.

14 L. Bruneau: Quantum entropic fluctuations and repeated interaction systems

Since the seminal works of Evans, Searles, Gallavotti and Cohen in the early 90's the study of entropic fluctuations has encountered a fast growing interest in the last decades, and many developments at least in classical systems. Its quantum counterpart however turned out to be very challenging. It has further been realized that the two time measurement protocol, introduced independently by Kurchan and Tasaki in 2000, sheds a new light on the problem. In this talk we will first introduce the problem of entropic fluctuations in quantum systems. In a second part we will concentrate on a specific class of models particularly suited to the problem, the so-called repeated interaction systems whose physical paradigm is the one-atom maser model.

This talk is based on a joint work with J.-F. Bougron.