Séminaire Probabilités et Statistique

It is a standard assumption that the Gaussian noises in stochastic systems are white in time and white space. This means that the noise at different point in space or in time are assumed to be uncorrelated. This leads to the Ito theory of integration. However, some time series data and other data indicate otherwise, some even exhibits long range dependence. In SDEs these imply that neither the Markov theory nor its martingale characterisation can be relied on. In SPDEs, the difficulty of irregularity coming from the white noise can be mitigated if they are replaced by smooth correlated noise. But other problems arrive. In this talk we shall explore these models and some phenomenons. New, as well as old, techniques in Stochastic Analysis will be explored.

After much success in using the double random current representation in the study of the Ising model, Duminil-Copin posed the question in 2016 of determining the (percolative) phase transition of the single random current. By relating the single random current to the loop O(1) model (also known as the high-temperature expansion and Eulerian percolation), we prove polynomial lower bounds for path probabilities (and infinite expectation of cluster sizes) for both the single random current and loop O(1) model corresponding to any supercritical Ising model on the hypercubic lattice. Thereby partially resolving the posed question.

In this talk, I will gently introduce graphical representations of the Ising model, their monotonicity properties and relations through Bernoulli sprinkling and the uniform even subgraph. Afterward, we discuss new results whose surprising proof takes inspiration from the toric code in quantum theory. Based on joint work with Ulrik Tinggaard Hansen and Boris Kjær: https://arxiv.org/abs/2306.05130.

Hawkes processes are a family of point processes for which the occurrence of any event increases the probability of further events occurring. Although the linear Hawkes process, for which a representation in the form of a superposition of branching processes exists, is particularly well studied, difficulties remain in estimating the parameters of the process from imperfect data (noisy, missing or aggregated data), since the usual estimation methods based on maximum likelihood or least squares do not necessarily offer theoretical guarantees or are numerically too costly.
In this work, we propose a spectral approach well-adapted to this context, for which we prove consistency and asymptotic normality. In order to derive these properties, we show that Hawkes processes can be studied through the scope of mixing, opening the use of central limit theorems that already exist in the literature.
I will then present two applications of this approach: to aggregated data (joint work with Gabriel Lang); and to noisy data (joint work with Anna Bonnet, Miguel Martinez and Maxime Sangnier).

Après avoir introduit les U-statistiques de données indépendantes, le théorème limite central fonctionnel et la loi des grands nombres correspondantes, nous présenterons des résultats récents pour des U-statistiques basées sur des suites stationnaires dépendantes à valeurs dans un espace métrique séparable. Nous traiterons le cas des suites beta-mélangeantes (qui se prêtent bien au couplage) ainsi que celui des U-statistiques à valeurs dans un espace de Hilbert. Ces dernières se révèlent utiles dans certains tests statistiques.

The stochastic dynamic matching problem has recently drawn attention in the stochastic-modeling community due to its numerous applications, ranging from supply-chain management to kidney exchange programs. In this presentation, we consider a matching problem in which items of different classes arrive according to independent Poisson processes, unmatched items are stored in a queue, and compatibility constraints are described by a simple graph on the classes, so that two items can be matched if their classes are neighbors in the graph. We analyze the efficiency of matching policies, not only in terms of system stability, but also in terms of matching rates between different classes. Our results rely on the observation that, under any stable policy, the matching rates satisfy a conservation equation that equates the arrival and departure rates of each item class.

This presentation is based on a joint work with Fabien Mathieu (LINCS) and Ana Bušić (Inria and PSL University). A preprint is available at the following address: https://arxiv.org/abs/2112.14457.

Imagine a collection of randomly distributed points in space, and build a graph on it according to some rules. We now take the sum of all edge lengths and want to know if this sum verifies a central limit theorem. In our context, the random collection of points is given by a Poisson measure and the sum of edge lengths is an example of a Poisson functional. In this talk, using the so-called Malliavin-Stein method, we show a result which allows us to derive central limit theorems, as well as speeds of convergence, for functionals of general Poisson measures, under minimal moment assumptions. Our main application is the closure of a conjecture about a convergence result of the Online Nearest Neighbour Graph.

Le Processus de saut renforcé par sommet (VRJP en raison de son acronyme anglais) est un processus aléatoire en temps continu auto-renforcé défini sur un graphe : plus il passe par un sommet, plus il est probable qu’il y revienne à l’avenir. Depuis les travaux de Sabot et Tarrès, on sait que le VRJP peut être vu comme un processus Markovien en environnement aléatoire. Cet environnement est caractérisé par un certain champ aléatoire U qui, étonnamment, était déjà connu par les physiciens en tant que modèle sigma super-symétrique H^{(2|2)}, un objet important en physique quantique. Par un changement de variable, on peut ramener l’étude du champ U à celle d’un autre champ beta. Ce champ beta permet de définir un opérateur de Schrödinger aléatoire H dont les propriétés sont reliées au VRJP. H est notamment inversible et son inverse G permet de retrouver l’environnement du VRJP. Dans cet exposé, nous expliquerons ce cheminement allant du VRJP vers les champs aléatoires et nous expliquerons les résultats connus sur ces champs et sur le VRJP. Pour finir, nous parlerons d’un résultat plus récent établissant la limite d’échelle de G sur le cercle. Cela permet de définir un opérateur de Schrödinger aléatoire continu analogue à H sur le cercle.

In the study of singular SPDEs, it has been a challenging problem to obtain a simple proof of a general probabilistic convergence result (BPHZ theorem). Differently from Chandra and Hairer’s Feynman diagram approach, Linares, Otto, Tempelmayr, and Tsatsoulis recently proposed an inductive proof based on the spectral gap inequality by using their multiindex language. Inspired by their approach, Hairer and Steele also obtained an inductive proof by using the regularity structure language. In this talk, we introduce an extension of the regularity structure including integrability exponents and provide a simpler proof of BPHZ theorem.
This talk is based on a joint work with Ismael Bailleul (Univ Brest).