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Rémi Catellier

Publications and source records attributed to Rémi Catellier.

8 recordsLinked to original sources
arXiv2024

Brownian particles controlled by their occupation measure

In this article, we study a finite horizon linear-quadratic stochastic control problem for Brownian particles, where the cost functions depend on the state and the occupation measure of the particles. To address this problem, we develop an Itô formula for the flow of occupation measure, which enables us to derive the associated Hamilton-Jacobi-Bellman equation. Then, thanks to a Feynman-Kac formula and the Boué-Dupuis formula, we construct an optimal strategy and an optimal trajectory. Finally, we illustrate our result when the cost-function is the volume of the sausage associated to the particles.

math.PR↗
arXiv2024

Regularization by noise for rough differential equations driven by Gaussian rough paths

We consider the rough differential equation with drift driven by a Gaussian geometric rough path. Under natural conditions on the rough path, namely non-determinism, and uniform ellipticity conditions on the diffusion coefficient, we prove path-by-path well-posedness of the equation for poorly regular drifts. In the case of the fractional Brownian motion $B^H$ for $H>\frac14$, we prove that the drift may be taken to be $κ>0$ Hölder continuous and bounded for $κ>\frac32 - \frac1{2H}$. A flow transform of the equation and Malliavin calculus for Gaussian rough paths are used to achieve such a result.

math.PR↗
arXiv2023

On the Robustness of Text Vectorizers

A fundamental issue in machine learning is the robustness of the model with respect to changes in the input. In natural language processing, models typically contain a first embedding layer, transforming a sequence of tokens into vector representations. While the robustness with respect to changes of continuous inputs is well-understood, the situation is less clear when considering discrete changes, for instance replacing a word by another in an input sentence. Our work formally proves that popular embedding schemes, such as concatenation, TF-IDF, and Paragraph Vector (a.k.a. doc2vec), exhibit robustness in the Hölder or Lipschitz sense with respect to the Hamming distance. We provide quantitative bounds for these schemes and demonstrate how the constants involved are affected by the length of the document. These findings are exemplified through a series of numerical examples.

cs.CL↗
arXiv2021

Pathwise regularization of the stochastic heat equation with multiplicative noise through irregular perturbation

Existence and uniqueness of solutions to the stochastic heat equation with multiplicative spatial noise is studied. In the spirit of pathwise regularization by noise, we show that a perturbation by a sufficiently irregular continuous path establish wellposedness of such equations, even when the drift and diffusion coefficients are given as generalized functions or distributions. In addition we prove regularity of the averaged field associated to a Lévy fractional stable motion, and use this as an example of a perturbation regularizing the multiplicative stochastic heat equation.

math.PR↗
arXiv2020

A mean-field approach to self-interacting network, convergence and regularity

The propagation of chaos property for a system of interacting particles, describing the spatial evolution of a network of interacting filaments is studied. The creation of a network of mycelium is analyzed as representative case, and the generality of the modeling choices are discussed. Convergence of the empirical density for the particle system to its mean field limit is proved, and a result of regularity for the solution is presented.

math.PR↗
arXiv2015

Rough linear transport equation with an irregular drift

We study the linear transport equation \[ \frac{\partial}{\partial t} u ( t,x ) +b ( t,x ) \cdot \nabla u ( t,x ) + \nabla u ( t,x ) \cdot \frac{\partial}{\partial t} X ( t ) =0, \hspace{2em} u ( 0,x ) =u_{0} ( x ) \] where $b$ is a vectorfield of limited regularity and $X$ a vector-valued Hölder continuous driving term. Using the theory of controlled rough paths we give a meaning to the weak formulation of the PDE and solve that equation for smooth vectorfields $b$. In the case of the fractional Brownian motion a phenomenon of regularization by noise is displayed.

math.PR↗
arXiv2011

The emergence of French statistics. How mathematics entered the world of statistics in France during the 1920s

This paper concerns the emergence of modern mathematical statistics in France after the First World War. Emile Borel's achievements are presented, and especially his creation of two institutions where mathematical statistics was developed: the {\it Statistical Institute of Paris University}, (ISUP) in 1922 and above all the {\it Henri Poincaré Institute} (IHP) in 1928. At the IHP, a new journal {\it Annales de l'Institut Henri Poincaré} was created in 1931. We discuss the first papers in that journal dealing with mathematical statistics.

math.HO↗