Bienvenue

Le LPSM est une unité mixte de recherche (UMR 8001) dépendant du CNRS, de Sorbonne Université et de l’Université Paris Cité. Le laboratoire compte environ 200 personnes (dont env. 90 permanents), répartis sur deux sites (Campus P. et M. Curie de Sorbonne Université et Campus Paris Rive Gauche de l’Université Paris Cité)

Les activités de recherche du LPSM couvrent un large spectre en Probabilités et Statistique, depuis les aspects les plus fondamentaux (qui incluent notamment l'Analyse Stochastique, la Géométrie Aléatoire, les Probabilités Numériques et les Systèmes Dynamiques) jusqu’aux applications à la Modélisation dans diverses disciplines (Physique, Biologie, Sciences des Données, Finance, Actuariat, etc), applications qui incluent des partenariats en dehors du monde académique.

Le LPSM est un laboratoire relativement récent. Cependant, ses composantes sont anciennes et proviennent du développement des « mathématiques du hasard » dans le centre de Paris, depuis le premier quart du 20ième siècle (voir ici pour plus de détails).

NB: Site largement inspiré de celui de l'IRIF (merci à eux pour la mise à disposition de leur maquette).

30.5.2023
Dominique Picard a été élue membre international de l'Académie des sciences américaine. Félicitations Dominique!

26.5.2023
Le projet présenté à l'ADEME par l'entreprise Califrais, et dans lequel le LPSM est partenaire, a reçu un financement pour 5 ans dans le cadre de l'appel d'offre Logistique 4.0 du PIA 4 “Stratégie d'accélération, Digitalisation et décarbonation des mobilités”.

19.9.2023
Piet Lammers est lauréat 2023-24 du prix et cours Claude-Antoine Peccot du Collège de France. Félicitations Piet!

26.5.2023
Quentin Berger, Claire Boyer et Max Fathi ont été nommés à l'Institut Universitaire de France lors de la campagne 2023. Félicitations à tous les trois!

4.6.2023
Conference Mathematics of disordered systems: a tribute to Francis Comets organized by Thierry Bodineau, Bernard Derrida, Giambattista Giacomin and Dasha Loukianova, Paris 5-7 June 2023.

toutes les anciennes actualités

(Ces actualités sont présentées selon un classement mêlant priorité et aléatoire.)

Séminaire de Probabilités
Mardi 3 octobre 2023, 14 heures, Jussieu, Salle Paul Lévy, 16-26 209
Guillaume Baverez (Université Humboldt de Berlin) Singular modules and null-vector equations in Liouville conformal field theory

In conformal field theory (CFT), the null-vector (or BPZ) equations are a set of PDEs satisfied by correlation functions (and conformal blocks) involving so-called “degenerate primary fields” (i.e. fields associated to degenerate representations of the Virasoro algebra). These equations are parametrised by a pair of positive integers (r,s) labelling the representation (“Kac table”), and by the topological type of the surface on which the CFT lives. The BPZ operator is a partial differential operator of order rs (the “level”) on the Teichmüller space of the surface, and it annihilates the conformal blocks if the corresponding representation is irreducible. In the probabilistic formulation of Liouville CFT, the BPZ equations have been shown to hold in some special cases (for correlation functions at level 2, on the sphere and to some extent in genus one), and they are key inputs in the proofs of certain exact formulae (e.g. the “DOZZ formula”). In this work, we generalise these results to all values of the parameters, and we make an explicit connection with the algebraic structure of the theory. Namely, we construct the degenerate modules for all (r,s) and show that they are irreducible. Then, we use a geometric characterisation of conformal blocks to translate this local information into a PDE on Teichmüller space. The talk will focus on the probabilistic aspects of this work: I will explain how to construct the modules using elementary properties of the Gaussian free field and Gaussian multiplicative chaos. An interesting feature of this construction is a probabilistic interpretation of the Kac table. Ongoing work with Baojun Wu.

Groupe de Travail Modélisation Stochastique
Mercredi 4 octobre 2023, 14 heures 15, Sophie Germain 1013
Tony Lelièvre (Ecole des Ponts ParisTech) Finding saddle points of energy landscapes: why and how?

The motivation of this presentation comes from the analysis of metastable stochastic process in statistical physics. One way to bridge the scale between full atomistic models and more coarse-grained descriptions is to use Markov State models parameterized by the Eyring Kramers formulas. These formulas give the hopping rates between local minima of the potential energy function. They require to identify the local minima and saddle points of the potential energy function. This approach is for example used in materials science (kinetic Monte Carlo models).

In this talk, I will first present a recent result obtained in collaboration with D. Le Peutrec (Université d'Orléans) and B. Nectoux (Université Clermont Auvergne) about the mathematical foundations of this approach, by deriving these Eyring-Kramers exit rates starting from the overdamped Langevin dynamics [1]. I will then introduce a recent algorithm we proposed together with P. Parpas (Imperial College London) in order to locate saddle points [2]. I will explain why these two works both rely on concentration properties of the eigenvectors of Witten Laplacians, in the small temperature regime.

References: [1] TL, D. Le Peutrec and B. Nectoux, Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary, https://arxiv.org/abs/2207.09284. [2] TL, P. Parpas /Using Witten Laplacians to locate index-1 saddle points/, https://arxiv.org/abs/2212.10135.

Séminaire de Probabilités
Mardi 10 octobre 2023, 14 heures, Jussieu, Salle Paul Lévy, 16-26 209
Quentin Berger (LPSM, Sorbonne Université) Limites d’échelles de systèmes désordonnés

Je présenterai quelques résultats récents concernant les limites d’échelles de systèmes désordonnés et des conséquences que l’on peu en tirer. Je me concentrerai essentiellement sur le modèle de Poland-Scheraga, aussi connu sous le nom de modèle d'accrochage, qui est utilisé pour décrire le phénomène de dénaturation de l’ADN : la question est de savoir si (et comment) le désordre perturbe la transition de dénaturation. Je décrirai notamment les résultats obtenus dans le cadre d’une version généralisée (censée être plus réaliste) du modèle, en collaboration avec Alexandre Legrand.

Séminaire de statistique
Mardi 10 octobre 2023, 9 heures 30, Jussieu en salle 15-16.201
Paul Escande On the Concentration of the Minimizers of Empirical Risks

Obtaining guarantees on the convergence of the minimizers of empirical risks to the ones of the true risk is a fundamental matter in statistical learning.

Instead of deriving guarantees on the usual estimation error, we will explore concentration inequalities on the distance between the sets of minimizers of the risks. We will argue that for a broad spectrum of estimation problems, there exists a regime where optimal concentration rates can be proven. The bounds will be showcased on a selection of estimation problems such as barycenters on metric space with positive or negative curvature, subspaces of covariance matrices, regression problems and entropic-Wasserstein barycenters.

Mathématiques financières et actuarielles, probabilités numériques
Jeudi 12 octobre 2023, 11 heures, Sophie Germain salle 1016
Céline Labart (Université de Savoie) Non encore annoncé.

Les probas du vendredi
Vendredi 13 octobre 2023, 11 heures, Jussieu, Salle Paul Lévy, 16-26 209
Justin Salez (Paris Dauphine) A new approach to the cutoff phenomenon

The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergoned by certain Markov processes in the limit where the number of states tends to infinity. Discovered forty years ago in the context of card shuffling, it has since then been established in a variety of contexts, including random walks on graphs and groups, high-temperature spin glasses, or interacting particle systems. Nevertheless, a general theory is still missing, and identifying the general mechanisms underlying this mysterious phenomenon remains one of the most fundamental problems in the area of mixing times. In this talk, I will give a self-contained introduction to this fascinating question, and then present a new approach based on entropy and curvature.

Séminaire de Probabilités
Mardi 17 octobre 2023, 14 heures, Jussieu, Salle Paul Lévy, 16-26 209
Roberto Imbuzeiro Oliveira (IMPA) Contact process over dynamical graphs

Mathématiques financières et actuarielles, probabilités numériques
Jeudi 19 octobre 2023, 11 heures, INRIA 2 Rue Simone Iff, 75012 Paris, France
Robert Denkert (HU Berlin) xtended Mean Field Control Problems with Singular Controls

We consider a novel class of extended mean field control (MFC) problems with singular controls, where the costs depend on the current state, control, and the joint law of the state-control-process. We prove an approximation of general singular controls with purely regular controls. Subsequently, we derive a dynamic programming principle and use it to establish a quasi-variational inequality (QVI) for the value function in the Wasserstein space. Finally, using the master equations of the approximating regular MFC problems, we establish a uniqueness result for our QVI characterisation of the value function of the MFC problem with singular controls. This presentation is based on joint work with Ulrich Horst.

Mathématiques financières et actuarielles, probabilités numériques
Jeudi 19 octobre 2023, 9 heures, Inria 2 Rue Simone Iff, 75012 Paris, France
Gudmund Pammer (ETH, Zurich) Stretched Brownian Motion: Analysis of a Fixed-Point Scheme

The fitting problem is a classical challenge in mathematical finance about finding martingales that satisfy specific marginal constraints. Building on the Bass solution to the Skorokhod embedding problem and optimal transport, Backhoff, Beiglböck, Huesmann, and Källblad propose a solution for the two-marginal problem: the stretched Brownian motion. Notably rich in structure, this process is an Ito diffusion and a continuous, strong Markov martingale. Following a similar approach, Conze and Henry-Larbordère recently introduced a novel local volatility model. This model, rooted in an extension of the Bass construction, is efficiently computable through a fixed-point scheme. In our presentation, we reveal the fixed-point scheme's intricate connection to the stretched Brownian motion and analyse its convergence. This presentation is based on joint work with Beatrice Acciaio and Antonio Marini.

Mathématiques financières et actuarielles, probabilités numériques
Jeudi 19 octobre 2023, 11 heures 45, Inria 2 Rue Simone Iff, 75012 Paris, France
Aurélien Alfonsi (ENPC) Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation.

We define and study convolution kernels that preserve nonnegativity. When the past dynamics of a process is integrated with a convolution kernel like in Stochastic Volterra Equations or in the jump intensity of Hawkes processes, this property allows to get the nonnegativity of the integral. We give characterizations of these kernels and show in particular that completely monotone kernels preserve nonnegativity. We then apply these results to analyze the stochastic invariance of a closed convex set by Stochastic Volterra Equations. We also get a comparison result in dimension one. Last, when the kernel is a positive linear combination of decaying exponential functions, we present a second order approximation scheme for the weak error that stays in the closed convex domain under suitable assumptions. We apply these results to the rough Heston model and give numerical illustrations.

Mathématiques financières et actuarielles, probabilités numériques
Jeudi 19 octobre 2023, 9 heures 45, Inria 2 Rue Simone Iff, 75012 Paris, France
Mehdi Talbi (LPSM) Sannikov’s contracting problem with many Agents

This work aims to study an extension of the celebrated Sannikov’s Principal-Agent problem to the multi-Agents case. In this framework, the contracts proposed by the Principal consist in a running payment, a retirement time and a final payment at retirement. After discussing how the Principal may derive optimal contracts in the N-Agents case, we explore the corresponding mean field model, with a continuous infinity of Agents. We then prove that the Principal’s problem can be reduced to a mixed control-and-stopping mean field problem, and we derive a semi-explicit solution of the first best contracting problem. This is a joint work with Thibaut Mastrolia and Nizar Touzi.

Les probas du vendredi
Vendredi 20 octobre 2023, 11 heures, Jussieu, Salle Paul Lévy, 16-26 209
Antoine Mouzard (ENS) Le générateur infinitésimal de la diffusion de Brox

Dans cet exposé, je vais présenter la diffusion de Brox. Il s'agit d'une modèle de dynamique aléatoire en milieu aléatoire, analogue continu de la marche aléatoire de Sinaï. Son générateur infinitésimal est un opérateur stochastique singulier qu'il est possible d'étudier à l'aide du calcul paracontrôlé. Après avoir introduit le contexte, je compte présenter sa construction dans le cadre d'un environnement périodique et les propriétés du processus que l'on peut en déduire.

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