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Nicolas Fournier

Publications and source records attributed to Nicolas Fournier.

At least 19 recordsLinked to original sources
arXiv2026

Asymptotics of a two-species particle system associated to the doubly parabolic Keller-Segel equation in the plane

We consider the two-species particle system introduced by Stevens (2000) related to the doubly parabolic Keller-Segel equation. It consists of $N$ cells and of a varying number of chemoattractant particles. Cells diffuse in the plane and follow the (mollified) empirical gradient of concentration of chemoattractant. Chemoattractant particles are produced by cells at some constant rate, diffuse and disappear at some constant rate. We show that when the sensitivity of cells to the chemoattractant is small enough, under some rather weak condition on the family of mollifiers, this system approximates the parabolic-parabolic Keller-Segel equation as $N\to \infty$. We also prove that when $N$ is fixed and when the production rate of chemoattractant particles tends to infinity, this system approximates the (non-Markovian) one-species system introduced Talay-Toma\v{s}evi\'c (2020) and further studied by the authors (2023).

math.PR↗
arXiv2026

Mean-field theory via dissociated arrays for particle systems interacting through noisy weights

We study a mean-field limit for a $N$-particle system in which each particle follows a diffusion and interacts with other particles through a weight on each directed edge. Each weight evolves according to its own nonlinear SDE driven by a Brownian motion, with coefficients involving the states of the two endpoint particles of the edge. The initial vertex and edge variables are assumed to have a dissociated Aldous--Hoover form. We construct the limiting nonlinear SDE by averaging the interaction over an independent neighbor and an edge input, prove its well-posedness, and show that the dissociated vertex-edge structure is propagated by the dynamics. This propagation property is an analogue of propagation of chaos in the case where the weight of each edge may remain correlated with the states of the two endpoint particles. Under either a bounded-observable assumption or a sub-Gaussian edge-input condition, the finite system converges to this limit through quantitative coupling estimates for a typical particle and a typical edge. We also prove the convergence of the empirical measure of particle's state pairs and their interaction weights.

math.PR↗
arXiv2025

Propagation of chaos for the homogeneous Boltzmann equation with moderately soft potentials

We show that the Kac particle system converges, as the number of particles tends to infinity, to the solution of the homogeneous Boltzmann equation, in the regime of moderately soft potentials, $\gamma \in (-2,0)$ with the common notation. This proves the propagation of chaos. We adapt the recent work of Imbert, Silvestre and Villani, to show that the Fisher information is nonincreasing in time along solutions to the Kac master equation. This estimate allows us to control the singularity of the interaction.

math.AP↗
arXiv2025

Fractional diffusion in convex domains and reflected isotropic stable processes

We establish the fractional diffusion limit of the kinetic scattering equation with diffusive boundary condition in a strongly convex bounded domain $\mathcal{D}\subset\mathbb{R}^d$. According to the nature of the boundary condition, two types of fractional heat equations may arise at the limit, corresponding to two types of isotropic stable processes reflected in $\mathcal{D}$. In both cases, when the process tries to jump across the boundary, it is stopped at the unique point where $\partial\mathcal{D}$ intersects the line segment defined by the attempted jump. It then leaves the boundary either continuously (for the first type) or by a power-law distributed jump (for the second type). The construction of these processes is done via an It\^o synthesis: we concatenate their excursions in the domain, which are obtained by translating, rotating and stopping the excursions of some stable processes reflected in the half-space. The key ingredient in this procedure is the construction of the boundary processes, i.e. the processes time-changed by their local time on the boundary, which solve stochastic differential equations driven by some Poisson measures of excursions. The well-posedness of these boundary processes relies on delicate estimates involving some geometric inequalities and the laws of the undershoot and overshoot of the excursion when it leaves the domain. We show that these reflected Markov processes are Markov and Feller, we study their infinitesimal generator and we write down the reflected fractional heat equations satisfied by their time-marginals.

math.PR↗
arXiv2025

On gelation for the Smoluchowski equation

Motivated by the recent results of Andreis-Iyer-Magnanini (2023), we provide a short proof, revisiting the one of Escobedo-Mischler-Perthame (2002), that for a large class of coagulation kernels, any weak solution to the Smoluchowski equation looses mass in finite time. The class of kernels we consider is essentially the same as the one of Andreis-Iyer-Magnanini (2023): homogeneous kernels of degree $\gamma>1$ not vanishing on the diagonal, or homogeneous kernels of degree $\gamma=1$ not vanishing on the diagonal with some additional logarithmic factor. We also show that when $\gamma=1$, the power of the logarithmic factor ensuring gelation may depend on the shape of the kernel.

math.AP↗
arXiv2023

Collisions of the supercritical Keller-Segel particle system

We study a particle system naturally associated to the $2$-dimensional Keller-Segel equation. It consists of $N$ Brownian particles in the plane, interacting through a binary attraction in $θ/(Nr)$, where $r$ stands for the distance between two particles. When the intensity $θ$ of this attraction is greater than $2$, this particle system explodes in finite time. We assume that $N>3θ$ and study in details what happens near explosion. There are two slightly different scenarios, depending on the values of $N$ and $θ$, here is one: at explosion, a cluster consisting of precisely $k_0$ particles emerges, for some deterministic $k_0\geq 7$ depending on $N$ and $θ$. Just before explosion, there are infinitely many $(k_0-1)$-ary collisions. There are also infinitely many $(k_0-2)$-ary collisions before each $(k_0-1)$-ary collision. And there are infinitely many binary collisions before each $(k_0-2)$-ary collision. Finally, collisions of subsets of $3,\dots,k_0-3$ particles never occur. The other scenario is similar except that there are no $(k_0-2)$-ary collisions.

math.PR↗
arXiv2022

Particle approximation of the doubly parabolic Keller-Segel equation in the plane

In this work, we study a stochastic system of N particles associated with the parabolic-parabolic Keller-Segel system. This particle system is singular and non Markovian in that its drift term depends on the past of the particles. When the sensitivity parameter is sufficiently small, we show that this particle system indeed exists for any $N\geq 2$, we show tightness in $N$ of its empirical measure, and that any weak limit point of this empirical measure, as $N\to\infty$, solves some nonlinear martingale problem, which in particular implies that its family of time-marginals solves the parabolic-parabolic Keller-Segel system in some weak sense. The main argument of the proof consists of a Markovianization of the interaction kernel: We show that, in some loose sense, the two-by-two path-dependant interaction can be controlled by a two-by-two Coulomb interaction, as in the parabolic-elliptic case.

math.PR↗
arXiv2021

A non-expanding transport distance for some structured equations

Structured equations are a standard modeling tool in mathematical biology. They areintegro-differential equations where the unknown depends on one or several variables, representing the state or phenotype of individuals. A large literature has been devoted to many aspects of these equations and in particular to the study of measure solutions.Here we introduce a transport distance closely related to the Monge-Kantorovich distance,which appears to be non-expanding for several (mainly linear) examples of structured equations.

math.AP↗
arXiv2021

Stability, well-posedness and regularity of the homogeneous Landau equation for hard potentials

We establish the well-posedness and some quantitative stability of the spatially homogeneous Landau equation for hard potentials, using some specific Monge-Kantorovich cost, assuming only that the initial condition is a probability measure with a finite moment of order $p$ for some $p>2$. As a consequence, we extend previous regularity results and show that all non-degenerate measure-valued solutions to the Landau equation, with a finite initial energy, immediately admit analytic densities with finite entropy. Along the way, we prove that the Landau equation instantaneously creates Gaussian moments. We also show existence of weak solutions under the only assumption of finite initial energy.

math.AP↗
arXiv2020

Simulated Annealing In $\mathbf{R}^d$ With Slowly Growing Potentials

We use a localization procedure to weaken the growth assumptions of Royer [8], Miclo [4] and Zitt [9] concerning the continuous-time simulated annealing in $\mathbf{R}^d$. We show that a transition occurs for potentials growing like $a \log \log |x|$ at infinity. We also study a class of potentials with possibly unbounded sets of local minima.

math.PR↗
arXiv2020

On The Simulated Annealing In $\mathbf{R}^d$

Using a localization procedure and the result of Holley-Kusuoka-Stroock [7] in the torus, we widely weaken the usual growth assumptions concerning the success of the continuous-time simulated annealing in $\mathbf{R}^d$. Our only assumption is the existence of an invariant probability measure for a sufficiently low temperature. We also prove, in an appendix, a non-explosion criterion for a class of time-inhomogeneous diffusions.

math.PR↗
arXiv2019

A coupling approach for the convergence to equilibrium for a collisionless gas

We use a probabilistic approach to study the rate of convergence to equilibrium for a collisionless (Knudsen) gas in dimension equal to or larger than 2. The use of a coupling between two stochastic processes allows us to extend and refine, in total variation distance, the polynomial rate of convergence given in [AG11] and [KLT13]. This is, to our knowledge, the first quantitative result in collisionless kinetic theory in dimension equal to or larger than 2 that does not require any symmetry of the domain, nor a monokinetic regime. Our study is also more general in terms of reflection at the boundary: we allow for rather general diffusive reflections and for a specular reflection component.

math-ph↗
arXiv2019

On a toy network of neurons interacting through their dendrites

Consider a large number $n$ of neurons, each being connected to approximately $N$ other ones, chosen at random. When a neuron spikes, which occurs randomly at some rate depending on its electric potential, its potential is set to a minimum value $v_{min}$, and this initiates, after a small delay, two fronts on the (linear) dendrites of all the neurons to which it is connected. Fronts move at constant speed. When two fronts (on the dendrite of the same neuron) collide, they annihilate. When a front hits the soma of a neuron, its potential is increased by a small value $w_n$. Between jumps, the potentials of the neurons are assumed to drift in $[v_{min},\infty)$, according to some well-posed ODE. We prove the existence and uniqueness of a heuristically derived mean-field limit of the system when $n,N \to \infty$ with $w_n \simeq N^{-1/2}$. We make use of some recent versions of the results of Deuschel and Zeitouni \cite{dz} concerning the size of the longest increasing subsequence of an i.i.d. collection of points in the plan. We also study, in a very particular case, a slightly different model where the neurons spike when their potential reach some maximum value $v_{max}$, and find an explicit formula for the (heuristic) mean-field limit.

math.PR↗
arXiv2019

Monge-Kantorovich distance for PDEs: the coupling method

We informally review a few PDEs for which the Monge-Kantorovich distance between pairs of solutions, possibly with some judicious cost function, decays: heat equation, Fokker-Planck equation, heat equation with varying coefficients, fractional heat equation with varying coefficients, homogeneous Boltzmann equation for Maxwell molecules, and some nonlinear integro-differential equations arising in neurosciences. We always use the same method, that consists in building a coupling between two solutions. This amounts to solve a well-chosen PDE posed on the Euclidian square of the physical space, i.e. doubling the variables. Finally, although the above method fails, we recall a simple idea to treat the case of the porous media equation. We also introduce another method based on the dual Monge-Kantorovich problem.

math.AP↗