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## Nos 50 dernières publications

• 17 January 2022

In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltoni-ans by adding potentials (functions depending only on the base point). This led to the definition of Mañé genericity: a property is generic if, given a Hamiltonian H, the set of potentials u such that H + u satisfies the property is generic. This notion is mostly used in the context of Hamiltonians which are convex in p, in the sense that ∂ 2 pp H is positive definite at each points. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians H + u and H/(1 − u) have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when H is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be perturbed by adding a small potential. This kind of question depend strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, Mañé perturbations of convex Hamiltonians. It is for example well-known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka-Smale theorem, see [5, 13]. There is no such result in the context of Hamiltonian vector fields, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [4, 2, 1]. The question was investigated only much more recently in the context of Mañé perturbations of convex Hamiltonians, see [9, 10]. It is proved in [10] that the same result holds : If H is a convex Hamiltonian and a is a regular value of H, then there exist arbitrarily small potentials u such that all periodic orbits (including iterated ones) of H + u at energy a are non-degenerate. The proof given in [10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work

Shahriar Aslani, Patrick Bernard

• 17 January 2022

We prove a bumpy metric theorem in the sense of Ma\~{n}e for non-convex Hamiltonians that are satisfying a certain geometric property.

Shahriar Aslani, Patrick Bernard

• 12 January 2022

We study a hard sphere gas at equilibrium, and prove that in the low density limit, the fluctuations converge to a Gaussian process governed by the fluctuating Boltzmann equation. This result holds for arbitrarily long times. The method of proof builds upon the weak convergence method introduced in the companion paper [8] which is improved by considering clusters of pseudo-trajectories as in [7].

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

• 11 January 2022

Throughout this PhD thesis we will study two probabilistic objects, Gaussian multiplicative chaos (GMC) measures and Liouville conformal field theory (LCFT). The GMC measures were first introduced by Kahane in 1985 and have grown into an extremely important field of probability theory and mathematical physics. Very recently GMC has been used to give a probabilistic definition of the correlation functions of LCFT, a theory that first appeared in Polyakov's 1981 seminal work, "Quantum geometry of bosonic strings". Once the connection between GMC and LCFT is established, one can hope to translate the techniques of conformal field theory in a probabilistic framework to perform exact computations on the GMC measures. We start from the BPZ equations for LCFT, introduced by Belavin, Polyakov and Zamolodchikov in 1983. The mechanism of these equations is studied in the last part of this thesis and we prove the higher order BPZ equations with a general formalism. Following the probabilistic methods established by Kupiainen-Rhodes-Vargas for the resolution of the BPZ equations and after overcoming several major difficulties, we obtain non trivial relations for some fundamental objects of LCFT. More precisely, we prove the exact formulas for all the four structure constants of LCFT on the disk with null cosmological constant in the bulk, one of which was solved by Remy in 2017. As a special case, we find the distribution of the total mass of GMC on the interval with log-singularities put on both ends, a conjecture that has been independently predicted by Ostrovsky and by Fyodorov, Le Doussal, and Rosso in 2009. Another direct consequence is the law of the total mass of GMC on the unit circle with a log-singularity, conjectured by Ostrovsky in 2016.

Tunan Zhu

• 8 January 2022

In this work, we prove the convergence of strong solutions of the Boltzman equation, for initial data having polynomial decay in the velocity variable, towards those of the incompressible Navier-Stokes-Fourier system. We show in particular that the solutions of the rescaled Boltzmann equation do not blow up before their hydrodynamic limit does. This is made possible by adapting the strategy from [7] of writing the solution to the Boltzmann equation as the sum a part with polynomial decay and a second one with Gaussian decay. The Gaussian part is treated with an approach reminiscent of the one from [17].

Pierre Gervais

• 30 December 2021

For dimensions $n \geq 3$, we classify singular solutions to the generalized Liouville equation $(-\Delta)^{n/2} u = e^{nu}$ on $\mathbb{R}^n \setminus \{0\}$ with the finite integral condition $\int_{\mathbb{R}^n} e^{nu} < \infty$ in terms of their behavior at $0$ and $\infty$. These solutions correspond to metrics of constant $Q$-curvature which are singular in the origin. Conversely, we give an optimal existence result for radial solutions. This extends some recent results on solutions with singularities of logarithmic type to allow for singularities of arbitrary order. As a key tool to the existence result, we derive a new weighted Moser--Trudinger inequality for radial functions.

Tobias König, Paul Laurain

• 17 December 2021

We prove an existence result for the principal-agent problem with adverse selection under general assumptions on preferences and allo-cation spaces. Instead of assuming that the allocation space is ﬁnite-dimensional or compact, we consider a more general coercivity condi-tion which takes into account the principal’s cost and the agents’ pref-erences. Our existence proof is simple and ﬂexible enough to adapt to partial participation models as well as to the case of type-dependent budget constraints.

Guillaume Carlier, Kelvin Shuangjian Zhang

• 9 December 2021

Hörmander's propagation of singularities theorem does not fully describe the propagation of singularities in subelliptic wave equations, due to the existence of doubly characteristic points. In the present paper, building upon a visionary conference paper by R. Melrose \cite{Mel86}, we prove that singularities of subelliptic wave equations only propagate along null-bicharacteristics and abnormal extremal lifts of singular curves, which are well-known curves in optimal control theory. We first revisit in depth the ideas sketched by R. Melrose in \cite{Mel86}, notably providing a full proof of its main statement. Making more explicit computations, we then explain how sub-Riemannian geometry and abnormal extremals come into play. This result shows that abnormal extremals have an important role in the classical-quantum correspondence between sub-Riemannian geometry and subelliptic operators. As a consequence, for $x\neq y$ and denoting by $K_G$ the wave kernel, we obtain that the singular support of the distribution $t\mapsto K_G(t,x,y)$ is included in the set of lengths of the normal geodesics joining $x$ and $y$, at least up to the time equal to the minimal length of a singular curve joining $x$ and $y$.

Cyril Letrouit

• 2 December 2021

We study the rapid decay property and polynomial growth for duals of bicrossed products coming from a matched pair of a discrete group and a compact group.

Pierre Fima, Hua Wang

• 30 November 2021

In this paper, we give an overview of the results established in [3] which provides the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres in 3D. In particular, we obtain a new system of hydrodynamic equations describing granular flows and prove existence of classical solutions to the aforementioned system. One of the main issue is to identify the correct relation between the restitution coefficient (which quantifies the rate of energy loss at the microscopic level) and the Knudsen number which allows us to obtain non trivial hydrodynamic behavior. In such a regime, we construct strong solutions to the inelastic Boltzmann equation, near thermal equilibrium whose role is played by the so-called homogeneous cooling state. We prove then the uniform exponential stability with respect to the Knudsen number of such solutions, using a spectral analysis of the linearized problem combined with technical a priori nonlinear estimates. Finally, we prove that such solutions converge, in a specific weak sense, towards some hydrodynamic limit that depends on time and space variables only through macroscopic quantities that satisfy a suitable modification of the incompressible Navier-Stokes-Fourier system.

Ricardo Alonso, Bertrand Lods, Isabelle Tristani

• 26 November 2021

We present an elementary approach to prove restriction theorems for particular surfaces for which the Tomas-Stein theorem does not apply, which in turn provide short proofs for well-known Strichartz estimates for associated PDEs. The method consists in applying simple restriction theorems to the level sets of the surfaces studied (for instance, spheres in the case of the cone) and then integrating the inequalities over all level sets. This allows for a different proof of sharp estimates for the wave equation, and of some estimates related to the Euler equations in the rotational framework.

Corentin Gentil, Côme Tabary

• 19 November 2021

We investigate the behavior of the spectrum of the continuous Anderson Hamiltonian $\mathcal{H}_L$, with white noise potential, on a segment whose size $L$ is sent to infinity. We zoom around energy levels $E$ either of order $1$ (Bulk regime) or of order $1\ll E \ll L$ (Crossover regime). We show that the point process of (appropriately rescaled) eigenvalues and centers of mass converge to a Poisson point process. We also prove exponential localization of the eigenfunctions at an explicit rate. In addition, we show that the eigenfunctions converge to well-identified limits: in the Crossover regime, these limits are universal. Combined with the results of our companion paper arXiv:2102.05393, this identifies completely the transition between the localized and delocalized phases of the spectrum of $\mathcal{H}_L$. The two main technical challenges are the proof of a two-points or Minami estimate, as well as an estimate on the convergence to equilibrium of a hypoelliptic diffusion, the proof of which relies on Malliavin calculus and the theory of hypocoercivity.

Laure Dumaz, Cyril Labbé

• 19 November 2021

Let $H\subset G$ be semisimple Lie groups, $\Gamma\subset G$ a lattice and $K$ a compact subgroup of $G$. For $n \in \mathbb N$, let $\mathcal O_n$ be the projection to $\Gamma \backslash G/K$ of a finite union of closed $H$-orbits in $\Gamma \backslash G$. In this very general context of homogeneous dynamics, we prove an equidistribution theorem for intersections of $\mathcal O_n$ with an analytic subvariety $S$ of $G/K$ of complementary dimension: if $\mathcal O_n$ is equidistributed in $\Gamma \backslash G/K$, then the signed intersection measure of $S \cap \mathcal O_n$ normalized by the volume of $\mathcal O_n$ converges to the restriction to $S$ of some $G$-invariant closed form on $G/K$. We give general tools to determine this closed form and compute it in some examples. As our main application, we prove that, if $\mathbb V$ is a polarized variation of Hodge structure of weight $2$ and Hodge numbers $(q,p,q)$ over a base $S$ of dimension $rq$, then the (non-exceptional) locus where the Picard rank is at least $r$ is equidistributed in $S$ with respect to the volume form $c_q^r$, where $c_q$ is the $q^{\textrm{th}}$ Chern form of the Hodge bundle. This generalizes a previous work of the first author which treated the case $q=r=1$. We also prove an equidistribution theorem for certain families of CM points in Shimura varieties, and another one for Hecke translates of a divisor in $\mathcal A_g$.

Salim Tayou, Nicolas Tholozan

• 18 November 2021

It has been known since Lanford [22] that the dynamics of a hard sphere gas is described in the low density limit by the Boltzmann equation, at least for short times. The classical strategy of proof fails for longer times, even close to equilibrium. In this paper, we introduce a weak convergence method coupled with a sampling argument to prove that the covariance of the fluctuation field around equilibrium is governed by the linearized Boltzmann equation globally in time (including in diffusive regimes). This method is much more robust and simple than the one devised in [4] which was specific to the 2D case.

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

• 15 November 2021

In this paper, we present a probabilistic study of rare phenomena of the cubic nonlinear Schrödinger equation on the torus in a weakly nonlinear setting. This equation has been used as a model to numerically study the formation of rogue waves in deep sea. Our results are twofold: first, we introduce a notion of criticality and prove a Large Deviations Principle (LDP) for the subcritical and critical cases. Second, we study the most likely initial conditions that lead to the formation of a rogue wave, from a theoretical and numerical point of view. Finally, we propose several open questions for future research.

Miguel Garrido, Ricardo Grande, Kristin Kurianski, Gigliola Staffilani

• 8 November 2021

This paper is dedicated to the study of a one-dimensional congestion model, consisting of two different phases. In the congested phase, the pressure is free and the dynamics is incompressible, whereas in the non-congested phase, the fluid obeys a pressureless compressible dynamics. We investigate the Cauchy problem for initial data which are small perturbations in the non-congested zone of travelling wave profiles. We prove two different results. First, we show that for arbitrarily large perturbations, the Cauchy problem is locally well-posed in weighted Sobolev spaces. The solution we obtain takes the form (vs, us)(t, x − x(t)), where x < x(t) is the congested zone and x > x(t) is the non-congested zone. The set {x = x(t)} is a free surface, whose evolution is coupled with the one of the solution. Second, we prove that if the initial perturbation is sufficiently small, then the solution is global. This stability result relies on coercivity properties of the linearized operator around a travelling wave, and on the introduction of a new unknown which satisfies better estimates than the original one. In this case, we also prove that travelling waves are asymptotically stable.

Anne-Laure Dalibard, Charlotte Perrin

• 5 November 2021

Let $F$ be a finite unramified extension of $\mathbb Q_p$ and $\bar\rho$ be an absolutely irreducible mod~$p$ $2$-dimensional representation of the absolute Galois group of $F$. Let $t$ be a tame inertial type of $F$. We conjecture that the deformation space parametrizing the potentially Barsotti--Tate liftings of $\bar\rho$ having type $t$ depends only on the Kisin variety attached to the situation, enriched with its canonical embedding into $(\mathbb P^1)^f$ and its shape stratification. We give evidences towards this conjecture by proving that the Kisin variety determines the cardinality of the set of common Serre weights $D(t,\bar\rho) = D(t) \cap D(\bar\rho)$. Besides, we prove that this dependance is nondecreasing (the smaller is the Kisin variety, the smaller is the number of common Serre weights) and compatible with products (if the Kisin variety splits as a product, so does the number of weights).

Xavier Caruso, Agnès David, Ariane Mézard

• 30 October 2021

Let $F$ be a finite unramified extension of $\mathbb Q_p$ and $\bar\rho$ be an absolutely irreducible mod~$p$ $2$-dimensional representation of the absolute Galois group of $F$. Let $t$ be a tame inertial type of $F$. We conjecture that the deformation space parametrizing the potentially Barsotti--Tate liftings of $\bar\rho$ having type $t$ depends only on the Kisin variety attached to the situation, enriched with its canonical embedding into $(\mathbb P^1)^f$ and its shape stratification. We give evidences towards this conjecture by proving that the Kisin variety determines the cardinality of the set of common Serre weights $D(t,\bar\rho) = D(t) \cap D(\bar\rho)$. Besides, we prove that this dependance is nondecreasing (the smaller is the Kisin variety, the smaller is the number of common Serre weights) and compatible with products (if the Kisin variety splits as a product, so does the number of weights).

Frédéric Jaeck, Angelo Guerraggio, Laurent Mazliak

• 30 October 2021

Let $F$ be a finite unramified extension of $\mathbb Q_p$ and $\bar\rho$ be an absolutely irreducible mod~$p$ $2$-dimensional representation of the absolute Galois group of $F$. Let $t$ be a tame inertial type of $F$. We conjecture that the deformation space parametrizing the potentially Barsotti--Tate liftings of $\bar\rho$ having type $t$ depends only on the Kisin variety attached to the situation, enriched with its canonical embedding into $(\mathbb P^1)^f$ and its shape stratification. We give evidences towards this conjecture by proving that the Kisin variety determines the cardinality of the set of common Serre weights $D(t,\bar\rho) = D(t) \cap D(\bar\rho)$. Besides, we prove that this dependance is nondecreasing (the smaller is the Kisin variety, the smaller is the number of common Serre weights) and compatible with products (if the Kisin variety splits as a product, so does the number of weights).

Frédéric Jaeck, Olivia Constantin

• 30 October 2021

As a generalization of the free semigroup algebras considered by Davidson and Pitts, and others, the second author and D.W. Kribs initiated a study of reflexive algebras associated with directed graphs. A free semigroupoid algebra L G \mathcal {L}_G is generated by a family of partial isometries, and initial projections, which act on a generalized Fock space spawned by the directed graph G G . We show that if the graph is finite, then L G \mathcal {L}_G is hyper-reflexive.

Frédéric Jaeck, Stephen Power

• 30 October 2021

As a generalization of the free semigroup algebras considered by Davidson and Pitts, and others, the second author and D.W. Kribs initiated a study of reflexive algebras associated with directed graphs. A free semigroupoid algebra L G \mathcal {L}_G is generated by a family of partial isometries, and initial projections, which act on a generalized Fock space spawned by the directed graph G G . We show that if the graph is finite, then L G \mathcal {L}_G is hyper-reflexive.

Frédéric Jaeck

• 29 October 2021

We address the problem of catching all speed $1$ geodesics of a Riemannian manifold with a moving ball: given a compact Riemannian manifold $(M,g)$ and small parameters $\e>0$ and $v>0$, is it possible to find $T>0$ and an absolutely continuous map $x:[0,T]\rightarrow M, t\mapsto x(t)$ satisfying $\|\dot{x}\|_{\infty}\leq v$ and such that any geodesic of $(M,g)$ traveled at speed $1$ meets the open ball $B_g(x(t),\e)\subset M$ within time $T$? Our main motivation comes from the control of the wave equation: our results show that the controllability of the wave equation can sometimes be improved by allowing the domain of control to move adequately, even very slowly. We first prove that, in any Riemannian manifold $(M,g)$ satisfying a geodesic recurrence condition (GRC), our problem has a positive answer for any $\varepsilon>0$ and $v>0$, and we give examples of Riemannian manifolds $(M,g)$ for which (GRC) is satisfied. Then, we build an explicit example of a domain $X\subset\R^2$ (with flat metric) containing convex obstacles, not satisfying (GRC), for which our problem has a negative answer if $\e$ and $v$ are small enough, i.e., no sufficiently small ball moving sufficiently slowly can catch all geodesics of $X$.

Cyril Letrouit

• 29 October 2021

In this work we construct an efficient numerical method to solve 3D Maxwell's equations in coaxial cables. Our strategy is based upon an hybrid explicit-implicit time discretization combined with edge elements on prisms and numerical quadrature. One of the objective is to validate numerically generalized Telegrapher's models that are used to simplify the 3D Maxwell equations into a 1D problem. This is the object of the second part of the article.

Akram Beni-Hamad, Geoffrey Beck, Sébastien Imperiale, Patrick Joly

• 27 October 2021

We study the norm of the two-dimensional Brownian motion conditioned to stay outside the unit disk at all times. We obtain sharp results on the rate of escape to infinity of the process of future minima. For this, we introduce a renewal structure attached to record times and values. Additional results are given for the long time behavior of the norm.

Orphée Collin, Francis Comets

• 11 October 2021

It is well-known that observability (and, by duality, controllability) of the elliptic wave equation, i.e., with a Riemannian Laplacian, in time $T_0$ is almost equivalent to the Geometric Control Condition (GCC), which stipulates that any geodesic ray meets the control set within time $T_0$. We show that in the subelliptic setting, GCC is never verified, and that subelliptic wave equations are never observable in finite time. More precisely, given any subelliptic Laplacian $\Delta=-\sum_{i=1}^m X_i^*X_i$ on a manifold $M$, and any measurable subset $\omega\subset M$ such that $M\backslash \omega$ contains in its interior a point $q$ with $[X_i,X_j](q)\notin \text{Span}(X_1,\ldots,X_m)$ for some $1\leq i,j\leq m$, we show that for any $T_0>0$, the wave equation with subelliptic Laplacian $\Delta$ is not observable on $\omega$ in time $T_0$. The proof is based on the construction of sequences of solutions of the wave equation concentrating on geodesics (for the associated sub-Riemannian distance) spending a long time in $M\backslash \omega$. As a counterpart, we prove a positive result of observability for the wave equation in the Heisenberg group, where the observation set is a well-chosen part of the phase space.

Cyril Letrouit

• 11 October 2021

In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schrödinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis.

Cyril Letrouit

• 10 October 2021

Tire sur l'élastique, la mathématique cherra. D'aucun verra dans l'allongement ou la contraction d'un chewing-gum des taux d'évolution, directs ou réciproques. Par exemple, +16 % dans un sens ou-6,25 % dans l'autre. D'aucun jouera les experts en stéganographie, en inscrivant un message privé sur un strap bien tendu, puis en le relâchant : devenu secret le texte se cache dans les plis ! D'aucun déduira l'aire de l'ellipse de demi-axes a et b, πab, de celle du disque de rayon a, πa 2 , d'une simple affinité. Mais il y a plus sérieux, comme le montreront les deux exemples ci-après abordables en classes de spécialité Mathématiques au lycée : la densité des rationnels parmi les réels d'une part, la propriété fondamentale du logarithme comme transformateur de produits en sommes.

Victor Rabiet, Karim Zayana, Ivan Boyer

• 7 October 2021

We introduce a random differential operator, that we call the $\mathtt{CS}_\tau$ operator, whose spectrum is given by the $\mbox{Sch}_\tau$ point process introduced by Kritchevski, Valk\'o and Vir\'ag (2012) and whose eigenvectors match with the description provided by Rifkind and Vir\'ag (2018). This operator acts on $\mathbf{R}^2$-valued functions from the interval $[0,1]$ and takes the form: $$2 \begin{pmatrix} 0 & -\partial_t \\ \partial_t & 0 \end{pmatrix} + \sqrt{\tau} \begin{pmatrix} d\mathcal{B} + \frac1{\sqrt 2} d\mathcal{W}_1 & \frac1{\sqrt 2} d\mathcal{W}_2\\ \frac1{\sqrt 2} d\mathcal{W}_2 & d\mathcal{B} - \frac1{\sqrt 2} d\mathcal{W}_1\end{pmatrix}\,,$$ where $d\mathcal{B}$, $d\mathcal{W}_1$ and $d\mathcal{W}_2$ are independent white noises. Then, we investigate the high part of the spectrum of the Anderson Hamiltonian $\mathcal{H}_L := -\partial_t^2 + dB$ on the segment $[0,L]$ with white noise potential $dB$, when $L\to\infty$. We show that the operator $\mathcal{H}_L$, recentred around energy levels $E \sim L/\tau$ and unitarily transformed, converges in law as $L\to\infty$ to $\mathtt{CS}_\tau$ in an appropriate sense. This allows to answer a conjecture of Rifkind and Vir\'ag (2018) on the behavior of the eigenvectors of $\mathcal{H}_L$. Our approach also explains how such an operator arises in the limit of $\mathcal{H}_L$. Finally we show that at higher energy levels, the Anderson Hamiltonian matches (asymptotically in $L$) with the unperturbed Laplacian $-\partial_t^2$. In a companion paper, it is shown that at energy levels much smaller than $L$, the spectrum is localized with Poisson statistics: the present paper therefore identifies the delocalized phase of the Anderson Hamiltonian.

Laure Dumaz, Cyril Labbé

• 5 October 2021

A cohomology class of a smooth complex variety of dimension n has coniveau at least c if it vanishes in the complement of a closed subvariety of codimension at least c, and has strong coniveau at least c if it comes by proper pushforward from the cohomology of a smooth variety of dimension at most n-c. We show that these two notions differ in general, both for integral classes on smooth projective varieties and for rational classes on smooth open varieties.

Olivier Benoist, John Christian Ottem

• 24 September 2021

Consider a weighted branching process generated by a point process on $[0,1]$, whose atoms sum up to one. Then the weights of all individuals in any given generation sum up to one, as well. We define a nested occupancy scheme in random environment as the sequence of balls-in-boxes schemes (with random probabilities) in which boxes of the $j$th level, $j=1,2,\ldots$ are identified with the $j$th generation individuals and the hitting probabilities of boxes are identified with the corresponding weights. The collection of balls is the same for all generations, and each ball starts at the root and moves along the tree of the weighted branching process according to the following rule: transition from a mother box to a daughter box occurs with probability given by the ratio of the daughter and mother weights. Assuming that there are $n$ balls, we give a full classification of regimes of the a.s.\ convergence for the number of occupied (ever hit) boxes in the $j$th level, properly normalized, as $n$ and $j=j_n$ grow to $\infty$. Here, $(j_n)_{n\in\mathbb N}$ is a sequence of positive numbers growing proportionally to $\log n$. We call such levels late, for the nested occupancy scheme gets extinct in the levels which grow super-logarithmically in $n$ in the sense that each occupied box contains one ball. Also, in some regimes we prove the strong laws of large numbers (a) for the number of the $j$th level boxes which contain at least $k$ balls, $k\geq 2$ and (b) under the assumption that the mean number of the first level boxes is finite, for the number of empty boxes in the $j$th level.

Alexander Iksanov, Bastien Mallein

• 16 September 2021

The aim of this thesis is to present new results in the analysis of metric measure spaces. We first extend to a certain class of spaces with doubling and Poincaré some weighted Sobolev inequalities introduced by V. Minerbe in 2009 in the context of Riemannian manifolds with non-negative Ricci curvature. In the context of RCD(0,N) spaces, we deduce a weighted Nash inequality and a uniform control of the associated weighted heat kernel. Then we prove Weyl’s law for compact RCD(K,N) spaces thanks to a pointwise convergence theorem for the heat kernels associated with a mGH-convergent sequence of RCD(K,N) spaces. Finally we address in the RCD(K,N) context a theorem from Bérard, Besson and Gallot which provides, by means of the heat kernel, an asymptotically isometric family of embeddings for a closed Riemannian manifold into its space of square integrable functions. We notably introduce the notions of RCD metrics, pull-back metrics, weak/strong convergence of RCD metrics, and we prove a convergence theorem analog to the one of Bérard, Besson and Gallot.

David Tewodrose

• 16 September 2021

We prove that for the Martinet wave equation with "flat" metric, which a subelliptic wave equation, singularities can propagate at any speed between 0 and 1 along any singular geodesic. This is in strong contrast with the usual propagation of singularities at speed 1 for wave equations with elliptic Laplacian.

Yves Colin de Verdière, Cyril Letrouit

• 9 September 2021

In this note we identify several mistakes appearing in the existing literature on non-stationary parametric bandits. More precisely, we study Generalized Linear Bandits (GLBs) in drifting environments, where the level of non-stationarity is characterized by a general metric known as the variation-budget. Existing methods to solve such problems typically involve forgetting mechanisms, which allow for a fine balance between the learning and tracking requirements of the problem. We uncover two significant mistakes in their theoretical analysis. The first arises when bounding the tracking error suffered by forgetting mechanisms. The second emerges when considering non-linear reward models, which requires extra care to balance the learning and tracking guarantees. We introduce a geometrical assumption on the arm set, sufficient to overcome the aforementioned technical gaps and recover minimax-optimality. We also share preliminary attempts at fixing those gaps under general configurations. Unfortunately, our solution yields degraded rates (w.r.t to the horizon), which raises new open questions regarding the optimality of forgetting mechanisms in non-stationary parametric bandits.

Louis Faury, Yoan Russac, Marc Abeille, Clément Calauzènes

• 7 September 2021

Il y a trois structures d’organisation interne explicitement revendiquées par les auteurs de l’Encylopédie : l’ordre alphabétique, la classification des articles dans le Système Figuré des Connaissances Humaines, et les renvois entre articles qui doivent indiquer la « liaison des matières ». En utilisant la version électronique de l’Encyclopédie réalisée par le laboratoire ARTFL de l’université de Chicago, c’est cette troisième structure, par nature même plus secrète, plus difficile à envisager dans son ensemble, que nous avons cherché à explorer. Un outil statistique simple nous permet de dresser la première « carte routière » de la structure de ces renvois. Bien que nécessairement imparfaite et à prendre avec précaution, cette carte révèle une grande cohérence de la structure sous-tendue par les renvois, dont nous montrons qu’elle est de nature assez différente de celle du Système Figuré. Notre analyse met également en lumière le statut particulier et ambigu de la catégorie Grammaire.

G. Blanchard, Mark Olsen

• 13 August 2021

Do physical processes compute? And what is a computation? These questions have gained a revival of interest in recent years, due to new technologies in physics, new ideas in computer sciences (for example quantum computing, networks, non-deterministic algorithms) and new concepts in logic. In this paper we examine a few directions, as well as the problems they bring to the surface.

Giuseppe Longo, Thierry Paul

• 2 August 2021

We study the Dyson-Ornstein-Uhlenbeck diffusion process, an evolving gas of interacting particles. Its invariant law is the beta Hermite ensemble of random matrix theory, a non-product log-concave distribution. We explore the convergence to equilibrium of this process for various distances or divergences, including total variation, entropy and Wasserstein. When the number of particles is sent to infinity, we show that a cutoff phenomenon occurs: the distance to equilibrium vanishes at a critical time. A remarkable feature is that this critical time is independent of the parameter beta that controls the strength of the interaction, in particular the result is identical in the non-interacting case, which is nothing but the Ornstein-Uhlenbeck process. We also provide a complete analysis of the non-interacting case that reveals some new phenomena. Our work relies among other ingredients on convexity and functional inequalities, exact solvability, exact Gaussian formulas, coupling arguments, stochastic calculus, variational formulas and contraction properties. This work leads, beyond the specific process that we study, to questions on the high-dimensional analysis of heat kernels of curved diffusions.

Jeanne Boursier, Djalil Chafai, Cyril Labbé

• 30 July 2021

On revoit les équations d'Einstein de la relativité générale dans le vide comme équations d'optimalité d'une sorte de problème de transport optimal quadratique généralisé.

Yann Brenier

• 28 July 2021

In this paper, we introduce and analyse numerical schemes for the homogeneous and the kinetic Lévy-Fokker-Planck equation. The discretizations are designed to preserve the main features of the continuous model such as conservation of mass, heavy-tailed equilibrium and (hypo)coercivity properties. We perform a thorough analysis of the numerical scheme and show exponential stability. Along the way, we introduce new tools of discrete functional analysis, such as discrete nonlocal Poincaré and interpolation inequalities adapted to fractional diffusion. Our theoretical findings are illustrated and complemented with numerical simulations.

Nathalie Ayi, Maxime Herda, Hélène Hivert, Isabelle Tristani

• 26 July 2021

In this paper, we study the Landau equation under the Navier-Stokes scaling in the torus for hard and moderately soft potentials. More precisely, we investigate the Cauchy theory in a perturbative framework and establish some new short time regularization estimates for our rescaled nonlinear Landau equation. These estimates are quantified in time and optimal, indeed, we obtain the instantaneous expected anisotropic gain of regularity (see [53] for the corresponding hypoelliptic estimates on the linearized Landau collision operator). Moreover, the estimates giving the gain of regularity in the velocity variable are uniform in the Knudsen number. Intertwining these new estimates on the Landau equation with estimates on the Navier-Stokes-Fourier system, we are then able to obtain a result of strong convergence towards this fluid system.

Kleber Carrapatoso, Mohamad Rachid, Isabelle Tristani

• 20 July 2021

We give necessary and sufficient conditions for the controllability of a Schr\"odinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups. This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schr\"odinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schr\"odinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main tools used in the proofs are (operator-valued) semi-classical measures constructed by use of representation theory and a notion of semi-classical wave packets that we introduce here in the context of groups of Heisenberg type.

Clotilde Fermanian Kammerer, Cyril Letrouit

• 24 June 2021

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups.

Miklos Abert, Nicolas Bergeron, Mikolaj Fraczyk, Damien Gaboriau

• 17 June 2021

We consider a subcritical Gaussian multiplicative chaos (GMC) measure defined on the unit interval $[0,1]$ and prove an exact formula for the fractional moments of the total mass of this measure. Our formula includes the case where log-singularities (also called insertion points) are added in $0$ and $1$, the most general case predicted by the Selberg integral. The idea to perform this computation is to introduce certain auxiliary functions resembling holomorphic observables of conformal field theory that will be solutions of hypergeometric equations. Solving these equations then provides nontrivial relations that completely determine the moments we wish to compute. We also include a detailed discussion of the so-called reflection coefficients appearing in tail expansions of GMC measures and in Liouville theory. Our theorem provides an exact value for one of these coefficients. Lastly, we mention some additional applications to small deviations for GMC measures, to the behavior of the maximum of the log-correlated field on the interval and to random hermitian matrices.

Guillaume Remy, Tunan Zhu

• 9 June 2021

We consider a subcritical Gaussian multiplicative chaos (GMC) measure defined on the unit interval $[0,1]$ and prove an exact formula for the fractional moments of the total mass of this measure. Our formula includes the case where log-singularities (also called insertion points) are added in $0$ and $1$, the most general case predicted by the Selberg integral. The idea to perform this computation is to introduce certain auxiliary functions resembling holomorphic observables of conformal field theory that will be solutions of hypergeometric equations. Solving these equations then provides nontrivial relations that completely determine the moments we wish to compute. We also include a detailed discussion of the so-called reflection coefficients appearing in tail expansions of GMC measures and in Liouville theory. Our theorem provides an exact value for one of these coefficients. Lastly, we mention some additional applications to small deviations for GMC measures, to the behavior of the maximum of the log-correlated field on the interval and to random hermitian matrices.

Bertrand Maury

• 7 June 2021

Let L and L' be two integral Euclidean lattices in the same genus. We give an asymptotic formula for the number of Kneser p-neighbors of L which are isometric to L', when the prime p goes to infinity. In the case L is unimodular, and if we fix furthermore a subgroup A ⊂ L, we also give an asymptotic formula for the number of p-neighbors of L containing A and which are isomorphic to L'. These statements explain numerical observations in [Che20] and [AC20]. In an Appendix, O. Taïbi shows how to deduce from Arthur's results the existence of global parameters associated to automorphic representations of definite orthogonal groups over the rationals.

Gaëtan Chenevier

• 7 June 2021

We characterize all random point measures which are in a certain sense stable under the action of branching. Denoting by $\circledast$ the branching convolution operation introduced by Bertoin and Mallein (2019), and by $\mathcal{Z}$ the law of a random point measure on the real line, we are interested in solutions to the fixed point equation $\mathcal E = \mathcal{Z} \circledast \mathcal E,$ with $\mathcal E$ a random point measure distribution. Under suitable assumptions, we characterize all solutions of this equation as shifted decorated Poisson point processes with a uniquely defined shift.

Pascal Maillard, Bastien Mallein

• 27 May 2021

This thesis studies the interaction between quantitative homogenization theory and two stochastic models: the supercritical percolation model and interacting particle systems. Stochastic homogenization focuses on large-scale properties in the random environment, and these two models represent generalization in the degenerate random environment and the dynamic random environment, respectively. In chapters 2 and 3, we study an efficient algorithm for the Dirichlet problem with random coefficients. This algorithm is proposed by Armstrong, Hannukainen, Kuusi and Mourrat, which allows an approximation in $H^1$ for the solution with high precision and low computational cost. We confirm its consistency in chapter 2, then extend it to the supercritical percolation model in chapter 3. Chapter 4 is devoted to the quantitative homogenization of the semigroup for the random walk on the infinite supercritical percolation cluster. Its probabilistic interpretation is as a quantitative local central limit theorem, and it also implies the convergence rate of the elliptic Green's function. The proof in this chapter combines several quantitative estimates on the percolation model: first-order correctors, flux, two-scaled expansion, and also the cluster density concentration. In chapters 5 and 6, we develop the homogenization theory for an interacting particle system without gradient condition in continuum configuration space. In chapter 5, we construct this model and prove its variance decay of Gaussian type. In chapter 6, we study its bulk diffusion coefficient and obtain a rate of convergence for the finite-volume approximation. Our strategy is the subadditivity-renormalization approach, and we also develop new functional inequalities adapted to this infinite-dimensional setting.

Chenlin Gu

• 19 May 2021

This thesis studies the interaction between quantitative homogenization theory and two stochastic models: the supercritical percolation model and interacting particle systems. Stochastic homogenization focuses on large-scale properties in the random environment, and these two models represent generalization in the degenerate random environment and the dynamic random environment, respectively. In chapters 2 and 3, we study an efficient algorithm for the Dirichlet problem with random coefficients. This algorithm is proposed by Armstrong, Hannukainen, Kuusi and Mourrat, which allows an approximation in $H^1$ for the solution with high precision and low computational cost. We confirm its consistency in chapter 2, then extend it to the supercritical percolation model in chapter 3. Chapter 4 is devoted to the quantitative homogenization of the semigroup for the random walk on the infinite supercritical percolation cluster. Its probabilistic interpretation is as a quantitative local central limit theorem, and it also implies the convergence rate of the elliptic Green's function. The proof in this chapter combines several quantitative estimates on the percolation model: first-order correctors, flux, two-scaled expansion, and also the cluster density concentration. In chapters 5 and 6, we develop the homogenization theory for an interacting particle system without gradient condition in continuum configuration space. In chapter 5, we construct this model and prove its variance decay of Gaussian type. In chapter 6, we study its bulk diffusion coefficient and obtain a rate of convergence for the finite-volume approximation. Our strategy is the subadditivity-renormalization approach, and we also develop new functional inequalities adapted to this infinite-dimensional setting.

Frédéric Jaeck

• 19 May 2021

This thesis studies the interaction between quantitative homogenization theory and two stochastic models: the supercritical percolation model and interacting particle systems. Stochastic homogenization focuses on large-scale properties in the random environment, and these two models represent generalization in the degenerate random environment and the dynamic random environment, respectively. In chapters 2 and 3, we study an efficient algorithm for the Dirichlet problem with random coefficients. This algorithm is proposed by Armstrong, Hannukainen, Kuusi and Mourrat, which allows an approximation in $H^1$ for the solution with high precision and low computational cost. We confirm its consistency in chapter 2, then extend it to the supercritical percolation model in chapter 3. Chapter 4 is devoted to the quantitative homogenization of the semigroup for the random walk on the infinite supercritical percolation cluster. Its probabilistic interpretation is as a quantitative local central limit theorem, and it also implies the convergence rate of the elliptic Green's function. The proof in this chapter combines several quantitative estimates on the percolation model: first-order correctors, flux, two-scaled expansion, and also the cluster density concentration. In chapters 5 and 6, we develop the homogenization theory for an interacting particle system without gradient condition in continuum configuration space. In chapter 5, we construct this model and prove its variance decay of Gaussian type. In chapter 6, we study its bulk diffusion coefficient and obtain a rate of convergence for the finite-volume approximation. Our strategy is the subadditivity-renormalization approach, and we also develop new functional inequalities adapted to this infinite-dimensional setting.

Frédéric Jaeck

• 19 May 2021

This thesis studies the interaction between quantitative homogenization theory and two stochastic models: the supercritical percolation model and interacting particle systems. Stochastic homogenization focuses on large-scale properties in the random environment, and these two models represent generalization in the degenerate random environment and the dynamic random environment, respectively. In chapters 2 and 3, we study an efficient algorithm for the Dirichlet problem with random coefficients. This algorithm is proposed by Armstrong, Hannukainen, Kuusi and Mourrat, which allows an approximation in $H^1$ for the solution with high precision and low computational cost. We confirm its consistency in chapter 2, then extend it to the supercritical percolation model in chapter 3. Chapter 4 is devoted to the quantitative homogenization of the semigroup for the random walk on the infinite supercritical percolation cluster. Its probabilistic interpretation is as a quantitative local central limit theorem, and it also implies the convergence rate of the elliptic Green's function. The proof in this chapter combines several quantitative estimates on the percolation model: first-order correctors, flux, two-scaled expansion, and also the cluster density concentration. In chapters 5 and 6, we develop the homogenization theory for an interacting particle system without gradient condition in continuum configuration space. In chapter 5, we construct this model and prove its variance decay of Gaussian type. In chapter 6, we study its bulk diffusion coefficient and obtain a rate of convergence for the finite-volume approximation. Our strategy is the subadditivity-renormalization approach, and we also develop new functional inequalities adapted to this infinite-dimensional setting.

Frédéric Jaeck, Laurent Mazliak, Emma Sallent del Colombo, Rossana Tazzioli

• 19 May 2021

This thesis studies the interaction between quantitative homogenization theory and two stochastic models: the supercritical percolation model and interacting particle systems. Stochastic homogenization focuses on large-scale properties in the random environment, and these two models represent generalization in the degenerate random environment and the dynamic random environment, respectively. In chapters 2 and 3, we study an efficient algorithm for the Dirichlet problem with random coefficients. This algorithm is proposed by Armstrong, Hannukainen, Kuusi and Mourrat, which allows an approximation in $H^1$ for the solution with high precision and low computational cost. We confirm its consistency in chapter 2, then extend it to the supercritical percolation model in chapter 3. Chapter 4 is devoted to the quantitative homogenization of the semigroup for the random walk on the infinite supercritical percolation cluster. Its probabilistic interpretation is as a quantitative local central limit theorem, and it also implies the convergence rate of the elliptic Green's function. The proof in this chapter combines several quantitative estimates on the percolation model: first-order correctors, flux, two-scaled expansion, and also the cluster density concentration. In chapters 5 and 6, we develop the homogenization theory for an interacting particle system without gradient condition in continuum configuration space. In chapter 5, we construct this model and prove its variance decay of Gaussian type. In chapter 6, we study its bulk diffusion coefficient and obtain a rate of convergence for the finite-volume approximation. Our strategy is the subadditivity-renormalization approach, and we also develop new functional inequalities adapted to this infinite-dimensional setting.

Frédéric Jaeck