arXiv subjects

Xavier Erny

Publications and source records attributed to Xavier Erny.

11 recordsLinked to original sources

arXiv2025

The first Fundamental Theorem of Calculus for functions defined on Wasserstein space

We establish an analogue of the first fundamental theorem of calculus for functions defined on the Wasserstein space of probability measures. Precisely, we show that if a function on the Wasserstein space is sufficiently regular in the sense of the linear functional derivative, then its integral is differentiable and the derivative coincides with the integrand. Our approach relies on a general differentiability criterion that connects the linear functional derivative, viewed as a Fr\'echet-derivative, and Dawson's weaker notion, which corresponds to a Gateaux-derivative. Under suitable regularity assumptions, it is possible to upgrade Gateaux-differentiability to Fr\'echet-differentiability in the infinite-dimensional setting of Wasserstein space.

math.FA↗

arXiv2024

Generators of measure-valued jump diffusions and convergence rate of diffusive mean-field models

The paper has two objectives: proving that the rate of convergence in distribution for mean-field models in CLT regime is $N^{-1/2}$, and obtaining explicit expressions for the infinitesimal generators of two types of measure-valued Markov processes (conditional law of McKean-Vlasov processes, and empirical measures of McKean-Vlasov systems). The proof of the convergence of mean-field system requires the second result about the generators, and both results need to study a notion of differentiability of measure-variable functions know as linear differentiability. Due to the particular framework that is studied, many technical difficulties arise compared to the existing literature. Two of the main problems are the following ones: the CLT regime implies that the limit measure-valued processes are not deterministic, and the empirical measure processes related to McKean-Vlasov equations with jumps are necessarily discontinuous. Both properties make the expressions of the generators more complicated than what is usually considered.

math.PR↗

arXiv2023

Annealed limit for a diffusive disordered mean-field model with random jumps

We study a sequence of $N-$particle mean-field systems, each driven by $N$ simple point processes $Z^{N,i}$ in a random environment. Each $Z^{N,i}$ has the same intensity $(f(X^N_{t-}))_t$ and at every jump time of $Z^{N,i},$ the process $X^N$ does a jump of height $U_i/\sqrt{N}$ where the $U_i$ are disordered centered random variables attached to each particle. We prove the convergence in distribution of $X^N$ to some limit process $\bar X$ that is solution to an SDE with a random environment given by a Gaussian variable, with a convergence speed for the finite-dimensional distributions. This Gaussian variable is created by a CLT as the limit of the patial sums of the $U_i.$ To prove this result, we use a coupling for the classical CLT relying on the result of [Komlós, Major and Tusnády (1976)], that allows to compare the conditional distributions of $X^N$ and $\bar X$ given the random environment, with the same Markovian technics as the ones used in [Erny, Löcherbach and Loukianova (2022)].

math.PR↗

arXiv2022

Sharp approximation and hitting times for stochastic invasion processes

We are interested in the invasion phase for stochastic processes with interactions when a single mutant with positive fitness arrives in a resident population at equilibrium. By a now classic approach, the first stage of the invasion is well approximated by a branching process. The macroscopic phase, when the mutant population is of the same order of the resident population, is described by the limiting dynamical system. We obtain sharper estimates and capture the intermediate mesoscopic phase for the invasive population. It allows us to characterize the hitting times of thresholds, which inherit a large variance from the first stages. These issues are motivated in particular by quantifying times to reach critical values for cancer population or epidemics.

math.PR↗

arXiv2022

Strong error bounds for the convergence to its mean field limit for systems of interacting neurons in a diffusive scaling

We consider the stochastic system of interacting neurons introduced in De Masi et al. (2015) and in Fournier and Löcherbach (2016) and then further studied in Erny, Löcherbach and Loukianova (2021) in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order $ 1 / \sqrt{N}.$ In between successive spikes, each neuron's potential follows a deterministic flow. In a previous article we proved the convergence of the system, as $N \to \infty$, to a limit nonlinear jumping stochastic differential equation. In the present article we complete this study by establishing a strong convergence result, stated with respect to an appropriate distance, with an explicit rate of convergence. The main technical ingredient of our proof is the coupling introduced in Komlós, Major and Tusnády (1976) of the point process representing the small jumps of the particle system with the limit Brownian motion

math.PR↗

arXiv2022

Mean field system of a two-layers neural model in a diffusive regime

We study a model of interacting neurons. The structure of this neural system is composed of two layers of neurons such that the neurons of the first layer send their spikes to the neurons of the second one: if $N$ is the number of neurons of the first layer, at each spiking time of the first layer, every neuron of both layers receives an amount of potential of the form $U/\sqrt{N},$ where $U$ is a centered random variable. This kind of structure of neurons can model a part of the structure of the visual cortex: the first layer represents the primary visual cortex V1 and the second one the visual area V2. The model consists of two stochastic processes, one modelling the membrane potential of the neurons of the first layer, and the other the membrane potential of the neurons of the second one. We prove the convergence of these processes as the number of neurons~$N$ goes to infinity and obtain a convergence speed. The proofs rely on similar arguments as those used in [Erny, Löcherbach, Loukianova (2022)]: the convergence speed of the semigroups of the processes is obtained from the convergence speed of their infinitesimal generators using a Trotter-Kato formula, and from the regularity of the limit semigroup.

math.PR↗

arXiv2022

Well-posedness and propagation of chaos for McKean-Vlasov equations with jumps and locally Lipschitz coefficients

We study McKean-Vlasov equations where the coefficients are locally Lipschitz continuous. We prove the strong well-posedness and a propagation of chaos property in this framework. These questions can be treated with classical arguments under the assumptions that the coefficients are globally Lipschitz continuous. In the locally Lipschitz case, we use truncation arguments and Osgood's lemma instead of Grönwall's lemma. This approach entails technical difficulties in the proofs, in particular for the existence of solution of the McKean-Vlasov equations that are considered. This proof relies on a Picard iteration scheme that is not guaranteed to converge in an $L^1-$sense because the coefficients are not Lipschitz continuous. However, we still manage to prove its convergence in distribution, and the (strong) well-posedness of the equation using a generalization of Yamada and Watanabe results.

math.PR↗

arXiv2021

White-noise driven conditional McKean-Vlasov limits for systems of particles with simultaneous and random jumps

We study the convergence of $N-$particle systems described by SDEs driven by Brownian motion and Poisson random measure, where the coefficients depend on the empirical measure of the system. Every particle jumps with a jump rate depending on its position and on the empirical measure of the system. Jumps are simultaneous, that is, at each jump time, all particles of the system are affected by this jump and receive a random jump height that is centred and scaled in $N^{-1/2}.$ This particular scaling implies that the limit of the empirical measures of the system is random, describing the conditional distribution of one particle in the limit system. We call such limits {\it conditional McKean-Vlasov limits}. The conditioning in the limit measure reflects the dependencies between coexisting particles in the limit system such that we are dealing with a {\it conditional propagation of chaos property}. As a consequence of the scaling in $N^{-1/2}$ and of the fact that the limit of the empirical measures is not deterministic the limit system turns out to be solution of a non-linear SDE, where not independent martingale measures and white noises appear having an intensity that depends on the conditional law of the process.

math.PR↗

arXiv2020

Mean field limits for interacting Hawkes processes in a diffusive regime

We consider a sequence of systems of Hawkes processes having mean field interactions in a diffusive regime. The stochastic intensity of each process is a solution of a stochastic differential equation driven by N independent Poisson random measures. We show that, as the number of interacting components N tends to infinity, this intensity converges in distribution in Skorohod space to a CIR-type diffusion. Moreover, we prove the convergence in distribution of the Hawkes processes to the limit point process having the limit diffusion as intensity. To prove the convergence results, we use analytical technics based on the convergence of the associated infinitesimal generators and Markovian semigroups.

math.PR↗

arXiv2020

A convergence criterion for systems of point processes from the convergence of their stochastic intensities

We study systems of simple point processes that admit stochastic intensities. We represent these point processes as thinnings of Poisson measures and are interested in a convergence result of such systems. This result states that, if the stochastic intensities of the limit point processes are independent of the underlying Poisson measures, the convergence in distribution in Skorohod topology of the stochastic intensities implies the same convergence for the point processes.

math.PR↗

arXiv2020

Conditional propagation of chaos for mean field systems of interacting neurons

We study the stochastic system of interacting neurons introduced in De Masi et al. (2015) and in Fournier and Löcherbach (2016) in a diffusive scaling. The system consists of $N$ neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to $0$ and all other neurons receive an additional amount of potential which is a centred random variable of order $ 1 / \sqrt{N}.$ In between successive spikes, each neuron's potential follows a deterministic flow. We prove the convergence of the system, as $N \to \infty$, to a limit nonlinear jumping stochastic differential equation driven by Poisson random measure and an additional Brownian motion $W$ which is created by the central limit theorem. This Brownian motion is underlying each particle's motion and induces a common noise factor for all neurons in the limit system. Conditionally on $W,$ the different neurons are independent in the limit system. This is the {\it conditional propagation of chaos} property. We prove the well-posedness of the limit equation by adapting the ideas of Graham (1992) to our frame. To prove the convergence in distribution of the finite system to the limit system, we introduce a new martingale problem that is well suited for our framework. The uniqueness of the limit is deduced from the exchangeability of the underlying system.

math.PR↗