Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

In this work, we study a diffusion process with jumps driven by a Hawkes process, where the intensity follows a piecewise deterministic Markov process. We begin by exploring the probabilistic properties of the system. Next, we focus on the nonparametric estimation of the coefficients in the associated stochastic differential equation (SDE), based on the minimization of various empirical contrast functions.

 Nous présentons la première méthode d'évaluation de la validité d'un clustering probabiliste applicable aux cadres paramétriques et non paramétriques. L'approche permet de tester la pertinence d'un clustering à partir des seules probabilités de classement estimées sur l'échantillon, à condition d'être capable d'échantillonner de nouvelles observations selon le modèle estimé.

In this talk, I will present an overview of recent results on finite-sample Probably Approximately Correct (PAC) and PAC-Bayesian bounds for learning partially observed dynamical systems in state-space form. For clarity, we begin with linear stochastic systems in discrete time, learned from a single trajectory, and then discuss extensions to more complex nonlinear settings.

The emergence of mean-curvature flow of an interface between different phases or populations is a phenomenon of long standing interest in statistical physics.  
In this talk, we review recent progress with respect to a class of reaction-diffusion stochastic particle systems on an $n$-dimensional lattice.
In such a process, particles can move across sites as well as be created/annihilated according to diffusion and reaction rates.  
These rates will be chosen so that there are two preferred particle mass density levels $a_1$, $a_2$ in `balance'.

Building on the presentation given at the colloquium on September 11, we go into some of the details of the homogenizations needed to obtain the sharp interface limit and associated fluctuations.

We study estimation and inference for the mean of real-valued random func- tions defined on a hypercube. The independent random functions are observed on a discrete, random subset of design points, possibly with heteroscedastic noise. We propose a novel optimal-rate estimator based on Fourier series expansions and establish a sharp non-asymptotic error bound in L2−norm.

In this seminar we present some recent results regarding density estimates for level sets of minimizers of local and nonlocal energies, which arise from phase separation problems, such as the Ginzburg-Landau energy functional. In particular, we prove density estimates when the phase separation is induced by a double-well potential which presents a slow growth from the pure phases. As a byproduct, we obtain the uniform convergence of the interfaces of any sequence of minimizers of a suitably rescaled energy functional to a set with Hausdorff codimension one.

The Iwatsuka Hamiltonian is a magnetic Schrödinger operator defined in the whole plane, whose magnetic field is constant in one direction. Its spectrum is purely absolutely continuous and has a band structure. These bands are generated by a family of real functions, called the band functions. This talk deals with the inverse problem of determining the Iwatsuka Hamiltonian by knowledge of its first band function. This is based on joint work with  M. Choulli (Lorraine), P. Hislop (Kentucky) et N. Kerraoui (Marseille).