We study stochastic multiple-fragmentation processes driven by a spatial flow. The final goal is actually to make a numerical simulation of the time evolution of a system of particles located on an Euclidean surface.

We take into account not only the fragmentation of the mass of a particle, but also of the kinetic energy. The talk is based on a joint work with Ioan R. Ionescu and Oana Lupascu-Stamate.

Considérons un graphe G_n uniformément choisi dans
l’ensemble des graphes à  n noeuds étiquetés avec des degrés D_1,…,D_n
donnés, eux-mêmes aléatoires i.i.d. tels que E[D^2]<∞ et P(D=2)E[D]. On se place ici dans le cas critique
E[D(D-1)]=E[D] et on suppose que P(D=k)∼ck^{-2-α}, 1<α<2. Des travaux de
Joseph 14, Riordan 12 et Conchon-Kerjan et Goldschmidt (à  paraître), il
résulte que le graphe G_n, après normalisation, converge en loi vers un
graphe continu aléatoire appelé graphe stable d'indice α. Nous
présenterons ici quelques propriétés géométriques de ce graphe limite.

Basé sur un travail en collaboration avec C. Goldschmidt et D.
Sénizergues.

To unveil the nature of 95% of the Universe, missions such as Euclid aim at reaching a few percent precision. In
this quest for precision, tensions between the standard cosmological model and observations already arise: local and global H0
measurements are incompatible at more than 3σ, anomalies emerge within the CMB, etc. These tensions suggest that we should perhaps not be so quickly inclined to disregard our observational site as a bias factor: Accuracy
is not Precision. Few percent precision and local-induced biases are of the same order of magnitude. A precise
mapping of the local distribution of matter is essential to properly account for these biases. Simulations constrained to resemble the local Universe constitute the tool of choice for such a mapping. I will summarize the genesis of the initial conditions of such simulations as well as present a few results that promise to tremendously impact our understanding of the local-induced biases that will matter in future analyses. Eventually, I will present the initial conditions of the
GMO-CLONES (GMO-Constrained LOcal & Nesting Environment Simulation) suite to reach an Accurate Precision Cosmology.

Un exemple de mouvement cinétique est celui d’une particule, soumise à 
des chocs aléatoires. En supposant que les chocs encodent
l’accélération, la vitesse suit une équation différentielle
stochastique, tandis que la position intègre simplement la vitesse. Le
mouvement résultant peut être assez délicat à  étudier, si par exemple la
particule est contrainte à  rester sur une surface, ou que la dynamique
de la vitesse est complexe. Je montrerai que sous des hypothèses très
simples de symétrie et d’ergodicité pour le processus vitesse, le
processus convenablement renormalisé converge vers un mouvement brownien
lorsque les chocs augmentent en intensité.

Bivariate signals appear in a broad range of applications: polarized waveforms in seismology and optics, current velocities in oceanography, etc. Formally, bivariate signals are 2D vector time series. Existing approaches for bivariate signal processing do not provide a straightforward description of the signal in terms of its polarization properties. For this purpose we introduce a new and generic framework based on a tailored quaternion Fourier transform.
This new framework re-establishes a clear interpretability in terms of polarization attributes of usual quantities such as spectral densities, linear filters, etc.
In this talk, I will introduce the main features of this approach, with the focus on second-order stationary random bivariate signals. I will discuss spectral analysis, linear filtering and some original decompositions of bivariate signals. Synthetic data will illustrate the usefulness of the proposed framework.

Pendant le séminaire, nous introduirons une formule d’intégration par partie non linéaire qui peut être vu comme une généralisation stochastique du lemme de Alekseev-Gröbner.

La preuve est basée sur le calcule de Malliavin et sur l’expression de certains intégrales stochastiques anticipatifs comme intégrales de Skorohod.

La formule que l’on présente induit une théorie de perturbations, i.e. une façon d’estimer, en terme de caractéristiques locales, l’erreur globale entre la solution exacte d’une équation différentielle stochastique et un processus d’Itô quelconque.

Si le temps le permet, nous parlerons des différences par rapport au résultat de perturbation établi précédemment par M. Hutzenthaler et A. Jentzen, et des applications comme la dérivation des taux de convergence en moyenne quadratique des schémas d’approximations pour ED(P)S.

(Travail en collaboration avec A. Hudde, M. Hutzenthaler, et A. Jentzen)

Integrating the increasing number of available multi-omics cancer data remains one of the main challenges to improve our understanding of cancer. Our approach is based on AMARETTO, an algorithm that integrates DNA methylation, DNA copy number and gene expression data to identify cancer driver genes and associates them to modules of co-expressed genes. We then propose a pancancer version of AMARETTO by connecting all modules in pancancer communities. This leads to the identification of major oncogenic pathways and master regulators involved in different cancers.

The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite time ruin probability in a standard risk model is greater or equal to the corresponding ruin probability evaluated in the associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T.
In this talk, we will consider the diffusion approximations of both the standard risk model and the associated risk model. We will prove that the associated model, when conveniently renormalized, converges in distribution to a gaussian process satisfying a simple SDE with explicit coefficients. We will then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model, which is well known. We will then conclude that the De Vylder and Goovaerts conjecture holds true for these diffusion limits.
This is a joint work with Stefan Ankirchner (University of Jena) and Christophette Blanchet-Scalliet (Ecole Centrale de Lyon and ICJ).

In a quasicrystal, the arrangement of the atoms is highly ordered (as
in an ordinary crystal) but non-periodic (unlike in a crystal). There
are various mathematical challenges in connection with quasicrystals.
From the point of view of statistical mechanics, the major open
problem is to provide a mathematical explanation of the formation and
stability of quasicrystals in presence of thermal fluctuations. In
this talk, I will present a (toy) lattice gas model with finite-range
interactions that has stable quasicrystal phases at positive
temperature (i.e., Gibbs measures supported at perturbations of
non-periodic tilings). The construction is based on old results on
cellular automata and tilings, in particular, a method of simulating
one cellular automaton with another that is resilient against noise,
and the existence of aperiodic sets of Wang tiles that are
deterministic in one direction.

The distribution of a Markov process with killing, conditioned to be
still alive at a given time, can be approximated by a Fleming-Viot
particle system. In such a system, each particle is simulated
independently according to the law of the underlying Markov process, and
branches onto another particle at each killing time. The purpose of this
talk is to present a central limit theorem for the law of the
Fleming-Viot particle system at a given time in the large population
limit. We will illustrate this result on an application in molecular
dynamics. This is a joint work with Frédéric Cérou, Bernard Delyon and
Mathias Rousset.

In the majority of solid cancers, secondary tumors (metastases) and associated complications are the main cause of death. In order to define the optimal therapeutic strategy for a given patient, one of the major current challenges is to estimate, at diagnosis, the burden of invisible metastases and how they will respond to treatments. In this talk, I will present research efforts towards the establishment of a predictive computational tool of metastatic development, with a particular emphasis on the assessment of mathematical models to empirical data (both experimental and clinical). I will first present the model’s framework, which is based on a physiologically-structured partial differential equation for the time dynamics of a population of metastases, combined to a nonlinear mixed-effects model for statistical representation of the distribution of the parameters in the population. Then, I will show results about the descriptive power of the model on data from clinically relevant ortho-surgical animal models of metastasis (breast and kidney tumors), with recent findings about differential effect of therapies between primary and secondary tumors. The talk will further be devoted to the translation of this modeling approach toward the clinical reality. Using clinical imaging data of brain metastasis from non-small cell lung cancer, several biological processes will be investigated to establish a minimal and biologically realistic model able to describe the data. Integration of this model into a biostatistical approach for individualized prediction of the model’s parameters from data only available at diagnosis will also be discussed. Together, these results represent a step forward towards the integration of mathematical modeling as a predictive tool for personalized medicine in oncology.

Nous présentons plusieurs algorithmes d’exploration de graphes aléatoires, markoviens dans le sens o๠leur implémentation est simultanée à  la construction-même du graphe par le modèle de configuration. Pour différents modèles, par des approximations fluides des processus markoviens sous-jacents, nous obtenons des estimations en grand graphe de (i) la taille de la famille indépendante maximale, avec des applications au protocole de télécommunication CSMA; (ii) la dynamique d’une épidémie de type SIR sur un réseau hétérogène et (iii) la taille d’un couplage maximal sur un grand graphe aléatoire, éventuellement orienté.

I will talk about the problem of detection of a change in mean in a Gaussian vector. A bump is a stepwise change within an interval of a given length but unknown location. We consider the problem of heterogeneous bump detection when the change occurs in mean and in variance of the observed vector and the detection of a bump for dependent observations. Minimax detectability conditions will be presented.
Joint work with A. Munk, M. Pohlmann and F. Werner.

Population dynamics of pathogens invading new territories continues to be of primary concern for both biologists and mathematicians. Extensive researches are mainly carried out throughout mathematical modeling to reconstruct the past dynamics of the alien species.We present a mechanistic-statistical approach that allows us to date and localize the invasion of an alien species and describe other epidemiological parameters as per example, the diffusion, the reproduction and the mortality parameter. The used approach is based on (i) a coupled reaction-diffusion-absorption sub-model that describes the dynamics of the epidemics in a heterogeneous domain and (ii) a stochastic sub-model that represents the observation process. Then, we will jointly estimate the initial conditions (date and site) and the epidemiological parameters using a Bayesian framework through an adaptive multiple importance sampling algorithm. We will show the results obtained in this framework on the basis of abundant post-introduction data gathered to draw up a surveillance plan on the expansion of Xylella fastidiosa, a phytopathogenic bacterium detected in South Corsica in 2015. Nevertheless, this approach could be applied to other post-emergent species in order to endorse a fast reaction.

The ovarian follicles are the basic anatomical and functional units of the ovaries, which are renewed from a
quiescent pool all along reproductive life. Follicular development involves a finely tuned sequence of growth and
maturation processes, involving complex cell dynamics. In their early stages of development, ovarian follicles are made up of a germ cell (oocyte), whose diameter increases steadily, and of surrounding proliferating somatic
cells, which are layered in a globally spherical and compact structure.
Here, we present two complementary modeling approaches dedicated to the first stages of a follicle develop-
ment, starting with the exit from the pool of quiescent (primordial) follicles leading to growth initiation, and
ending up just before the breaking of the spherical symmetry induced by the follicle cavitation (formation of
the antrum cavity).
The initiation phase is described by joint stochastic dynamics accounting for cell shape transitions (from
a flattened to a cuboidal shape) and proliferation of reshaped cells. We can derive the mean time elapsed before all cells have changed shapes and the corresponding increment in the total cell number, which is fitted
to experimental data retrieved from primordial follicles (single layered follicle with only flattened cells) and primary follicles (single layered follicles with only cuboidal cells).
The next stages, characterized by the accumulation of cell layers around the oocyte, are described by
multi-type structured models in either a stochastic or deterministic framework. We have designed a linear age-structured stochastic (Bellman-Harris branching) process ruling the changes in the number of follicular cells and their distribution into successive layers, which is inspired from the nonlinear model initially introduced in [1], as well as is deterministic counterpart (multi-dimensional Mc Kendrick Von Foerster). We have studied the large-time behavior of the models and derived explicit analytical formulas characterizing an exponential growth
of the population (Malthus parameter, asymptotic cell number moments and stable age distribution). We have
compared the theoretical and numerical outputs of the models with experimental biological data informing on follicle morphology in the ovine species (follicle and oocyte diameters, layer number and total cell number) from the primary to the pre-antral stage. In addition, in the case of age independent division rates, we have established the structural identifiability of the parameters, and estimated the parameter values fitting the cell numbers in each layer during the early stages of follicle development.

[1 ] Clément F., Michel P., Monniaux D., Stiehl T., Coupled somatic cell kinetics and germ cell growth:
mutliscale model-based insight on ovarian follicular development,
Multiscale et Modeling & Simulation
, 11(3), 719-746, 2013.

[2 ] Clément F., Robin F., Yvinec R., Analysis and calibration of a linear model for structured cell populations with unidirectional motion : Application to the morphogenesis of ovarian follicles,
Submitted. https://arxiv.or/abs/1712.05372

Nous considérons des systèmes de processus de Hawkes structurés en un nombre fini de populations, avec des interactions du type champ moyen. Ces processus décrivent les instants de décharge électrique de neurones en interactions. Nous décrivons la limite lorsque le nombre de neurones par population tend vers l’infini et nous montrons que dans certains cas, le processus limite possède des solutions périodiques qui correspondent aux orbites périodiques d’un système dynamique associé au processus limite. Nous établissons aussi des théorèmes limites qui décrivent la convergence du système de neurones à  taille finie vers ces orbites périodiques.

Ces résultats sont basés sur un travail en collaboration avec S. Ditlevsen.

In this talk, we present some inference methods in some multivariate models with
multiple unknown change-points when the target parameter is suspected to satisfy an uncertain constraint. We waive the assumptions on the error terms and establish the joint asymptotic normality of the unrestricted estimator and the restricted estimator. Further, we propose a class of shrinkage estimators that includes as a special case the unrestricted estimator, the estimator restricted as well as James-Stein type estimators. To study the performance of the proposed estimators, we generalize some classical identities underlying the multivariate Gaussian random samples or, more generally, the multivariate elliptically contoured random samples. Finally, we prove that shrinkage estimators dominate the unrestricted estimator.

Les populations composées de deux types d’individus (ou phénotypes) peuvent naturellement s’étudier grâce aux marches aléatoires planaires. Dans cet exposé je ferai un survol des techniques existantes et présenterai également de nouvelles idées pour l’étude de telles populations bidimensionnelles. Nous commencerons par le cas des populations homogènes, qui correspondent à  des modèles maintenant classiques de marches aléatoires dans le quart de plan. Nous présenterons ensuite une classe de marches aléatoires inhomogènes ayant une interprétation biologique naturelle. En passant nous lierons ces marches aléatoires inhomogènes avec d’autres processus, provenant de la théorie des files d’attente (joindre la file la plus courte), de processus de branchement, de modèles d’urnes et de théorie du potentiel. Il s’agit de différents travaux en commun avec Gerold Alsmeyer (Mà¼nster), Irina Kurkova (Paris 6), Pauline Lafitte (ECP) et Chi Tran (Lille).

Un défaut habituel des algorithmes de Monte Carlo par chaînes de Markov
est leur difficulté à  explorer l’espace convenablement, ce qui mène
à  de grandes erreurs d’estimation. L’algorithme de « métadynamique »,
introduit par Bussi, Laio et Parrinello dans les années 2000 illustre l’une des stratégies possibles pour contourner la difficulté, en gardant la mémoire de la trajectoire passée et en l’utilisant pour biaiser le processus et le pousser vers des régions peu visitées.
L’analyse des processus sous-jacent n’est pas aisée en général ; nous discuterons d’un modèle jouet, que l’on peut traiter par des outils de la littérature des processus auto-répulsifs.

Travail en commun avec B. Jourdain et T. Lelièvre (Ecole des Ponts ParisTech).

On essaye d’étudier une classe d’algorithmes du type Monte Carlo Tree Search pour des jeux
déterministes à  trait alterné et information complète (du type morpion, puissance 4, etc.).