arXiv subjects

Eva Löcherbach

Publications and source records attributed to Eva Löcherbach.

At least 19 recordsLinked to original sources

arXiv2026

Weak conditional propagation of chaos for systems of interacting particles with nearly stable jumps

We consider a system of $N$ interacting particles, described by SDEs driven by Poisson random measures, where the coefficients depend on the empirical measure of the system. Every particle jumps with a jump rate depending on its position. When this happens, all the other particles of the system receive a small random kick which is distributed according to a heavy-tailed random variable belonging to the domain of attraction of an $α$-stable law and scaled by $N^{-1/α},$ where $0<α<2$. We call these jumps collateral jumps. Moreover, in case $0<α<1$, the jumping particle itself undergoes a macroscopic, main jump. Such systems appear in the modeling of large neural networks, such as the human brain. Using a representation of the collateral jump sum as a time-changed random walk, we prove the convergence in law, in Skorokhod space, of this system to a limit infinite-exchangeable system of SDEs driven by a common stable process. This stable process arises due to the stable central limit theorem, and the particles in the limit system are independent and identically distributed, conditionally on that. That is, the $N$-particle system exhibits the conditional propagation of chaos property.

math.PR↗

arXiv2026

Nonparametric estimation of the jump rate in mean field interacting systems of neurons

We consider finite systems of $N$ interacting neurons described by non-linear Hawkes processes in a mean field frame. Neurons are described by their membrane potential. They spike randomly, at a rate depending on their potential. In between successive spikes, their membrane potential follows a deterministic flow. We estimate the spiking rate function based on the observation of the system of $N$ neurons over a fixed time interval $[0,t]$. Asymptotic are taken as $N,$ the number of neurons, tends to infinity. We introduce a kernel estimator of Nadaraya-Watson type and discuss its asymptotic properties with help of the deterministic dynamical system describing the mean field limit. We compute the minimax rate of convergence in an $L^2 -$error loss over a range of Hölder classes and obtain the classical rate of convergence $ N^{ - 2β/ ( 2 β+ 1)} , $ where $ β$ is the regularity of the unknown spiking rate function.

math.PR↗

arXiv2026

On large networks of integrate-and-fire neurons with short-term synaptic plasticity

This work studies the mean-field limit of large networks of interacting stochastic leaky integrate-and-fire (LIF) neurons subject to short-term synaptic depression (STD). The macroscopic dynamics of this system is governed by a two-dimensional, non-linear McKean-Vlasov equation that couples the evolution of the neurons' membrane potentials with a synaptic depression variable. We investigate the long-time behavior of this limit system. To this end, we introduce an auxiliary linearized Markov process by freezing the interaction non-linearity to a constant. By exploiting the regeneration of the membrane potential at spike times, we are able to explicitly compute the conditional expectation of the synaptic depression variable, conditionally on the potential value, under the invariant measure of this two-dimensional linear process. This is a crucial ingredient to study time-dependent local perturbations thereof. As a consequence we are able to identify an analytic criterion guaranteeing the local stability of any invariant probability measure of the fully non-linear system. This stability criterion is formulated in terms of the zeros of the Laplace transform of a specific linear response function. Finally, we provide numerical examples demonstrating that the two-dimensional framework induces a richer spectrum of long-time dynamics than purely one-dimensional models. For example, synaptic depression can lead to low-frequency oscillations around a unique, unstable invariant measure where the oscillations are much slower than the neurons' firing rates.

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arXiv2026

Emergence of Common Noise: Quantitative Conditional Propagation of Chaos

We study a dynamical invariance principle for interacting particle systems with mean-field interactions and common noise emerging through a collective stochastic perturbation. The particle dynamics combine autonomous evolution with a weakly scaled random bombardment whose cumulative effect generates a Brownian common noise in the large population limit. Working in a general abstract framework based on stochastic flows and the stochastic sewing lemma, we establish quantitative conditional propagation of chaos estimates for both discrete Euler schemes and their continuous-time flow limits. Our approach yields explicit Wasserstein convergence rates and applies in particular to jump-diffusion models motivated by interacting spiking neuron systems. The analysis relies on quantitative central limit theorems of Rio and Bonis together with a stochastic sewing argument adapted to $L^2$-valued flows.

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arXiv2025

A new look at perfect simulation for chains with infinite memory

In this article we introduce two new perfect simulation algorithms for chains with infinite memory. Both algorithms belong to the coupling of past procedures. The novelty of our approach is that it allows to include unknown states to the possible past symbols such that we can also deal with sparsely distributed past dependencies. In our first algorithm, spontaneous occurrence of symbols is possible. This means that there is a positive probability that the chain chooses the next symbol independently of the past. Our second algorithm deals with the case in which spontaneous occurrence of symbols is not possible. Chains with infinite memory are discrete-time stochastic processes in which the distribution of the next symbol depends on all past symbols. These transition probabilities are described by a probability kernel. Our results give conditions on the way the dependency of the transition kernel on long past strings decays, guaranteeing that our algorithms stop after a finite number of steps almost surely. Strengthening these conditions, we show that the mean number of steps of our algorithms is finite. We discuss the consequence of having a coupling from the past algorithm with such properties and we present examples in which our results can be applied while others result in the literature cannot be applied.

math.PR↗

arXiv2025

Inferring the dependence graph density of binary graphical models in high dimension

We consider a system of binary interacting chains describing the dynamics of a group of $N$ components that, at each time unit, either send some signal to the others or remain silent otherwise. The interactions among the chains are encoded by a directed Erdös-Rényi random graph with unknown parameter $ p \in (0, 1) .$ Moreover, the system is structured within two populations (excitatory chains versus inhibitory ones) which are coupled via a mean field interaction on the underlying Erdös-Rényi graph. In this paper, we address the question of inferring the connectivity parameter $p$ based only on the observation of the interacting chains over $T$ time units. In our main result, we show that the connectivity parameter $p$ can be estimated with rate $N^{-1/2}+N^{1/2}/T+(\log(T)/T)^{1/2}$ through an easy-to-compute estimator. Our analysis relies on a precise study of the spatio-temporal decay of correlations of the interacting chains. This is done through the study of coalescing random walks defining a backward regeneration representation of the system. Interestingly, we also show that this backward regeneration representation allows us to perfectly sample the system of interacting chains (conditionally on each realization of the underlying Erdös-Rényi graph) from its stationary distribution. These probabilistic results have an interest in its own.

math.ST↗

arXiv2025

Strong propagation of chaos for systems of interacting particles with nearly stable jumps

We consider a system of $N$ interacting particles, described by SDEs driven by Poisson random measures, where the coefficients depend on the empirical measure of the system. Every particle jumps with a jump rate depending on its position. When this happens, all the other particles of the system receive a small random kick which is distributed according to a heavy tailed random variable belonging to the domain of attraction of an $α-$ stable law and scaled by $N^{-1/α},$ where $0 < α<2 .$ We call these jumps collateral jumps. Moreover, in case $ 0 < α< 1, $ the jumping particle itself undergoes a macroscopic, main jump. Such systems appear in the modeling of large neural networks, such as the human brain. The particular scaling of the collateral jumps implies that the limit of the empirical measures of the system is random and equals the conditional distribution of one typical particle in the limit system, given the source of common noise. Thus the system exhibits the conditional propagation of chaos property. The limit system turns out to be solution of a non-linear SDE, driven by an $ α-$stable process. We prove strong unique existence of the limit system and introduce a suitable coupling to obtain the strong convergence of the finite to the limit system, together with precise error bounds for finite time marginals.

math.PR↗

arXiv2024

Mean field limits of interacting particle systems with positive stable jumps

This note is a companion article to the recent paper Löcherbach, Loukianova, Marini (2024). We consider mean field systems of interacting particles. Each particle jumps with a jump rate depending on its position. When jumping, a macroscopic quantity is added to its own position. Moreover, simultaneously, all other particles of the system receive a small random kick which is distributed according to a positive $α-$stable law and scaled in $N^{-1/α},$ where $0 < α< 1.$ In between successive jumps of the system, the particles follow a deterministic flow with drift depending on their position and on the empirical measure of the total system. In a more general framework where jumps and state space do not need to be positive, we have shown in Löcherbach, Loukianova, Marini (2024) that the mean field limit of this system is a McKean-Vlasov type process which is solution of a non-linear SDE, driven by an $ α-$stable process. Moreover we have obtained in Löcherbach, Loukianova, Marini (2024) an upper bound for the strong rate of convergence with respect to some specific distance disregarding big jumps of the limit stable process. In the present note we consider the specific situation where all jumps are positive and particles take values in $[ 0, + \infty [ . $ We show that in this case it is possible to improve upon the error bounds obtained in Löcherbach, Loukianova, Marini (2024) by using an adhoc distance obtained after applying a concave space transform to the trajectories. The distance we propose here takes into account the big jumps of the limit $ α-$stable subordinator.

math.PR↗

arXiv2024

Propagation of chaos and phase transition in a stochastic model for a social network

We consider a model for a social network with N interacting social actors. This model is a system of interacting marked point processes in which each point process indicates the successive times in which a social actor expresses a "favorable" (+1) or "contrary" (-1) opinion. The orientation and the rate at which an actor expresses an opinion is influenced by the social pressure exerted on this actor. The social pressure of an actor is reset to 0 when the actor expresses an opinion, and simultaneously the social pressures on all the other actors change by h/N in the direction of the opinion that was just expressed. We prove propagation of chaos of the system, as N diverges to infinity, to a limit nonlinear jumping stochastic differential equation. Moreover, we prove that under certain conditions the limit system exhibits a phase transition described as follows. If h is smaller or equal than a certain threshold, the limit system has only the null Dirac measure as an invariant probability measure, corresponding to a vanishing social pressure on all actors. However, if h is greater than the threshold, the system has two additional non-trivial invariant probability measures. One of these measures has support on the positive real numbers and the other is obtained by symmetrization with respect to 0, having thus support on the negative real numbers.

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

On a finite-size neuronal population equation

Population equations for infinitely large networks of spiking neurons have a long tradition in theoretical neuroscience. In this work, we analyze a recent generalization of these equations to populations of finite size, which takes the form of a nonlinear stochastic integral equation. We prove that, in the case of leaky integrate-and-fire (LIF) neurons with escape noise and for a slightly simplified version of the model, the equation is well-posed and stable in the sense of Brémaud-Massoulié. The proof combines methods from Markov processes taking values in the space of positive measures and nonlinear Hawkes processes. For applications, we also provide efficient simulation algorithms.

math.PR↗

arXiv2022

Kalikow decomposition for counting processes with stochastic intensity and application to simulation algorithms

We propose a new Kalikow decomposition for continuous time multivariate counting processes, on potentially infinite networks. We prove the existence of such a decomposition in various cases. This decomposition allows us to derive simulation algorithms that hold either for stationary processes with potentially infinite network but bounded intensities, or for processes with unbounded intensities in a finite network and with empty past before 0. The Kalikow decomposition is not unique and we discuss the choice of the decomposition in terms of algorithmic efficiency in certain cases. We apply these methods on several examples: linear Hawkes process, age dependent Hawkes process, exponential Hawkes process, Galves-Löcherbach process.

math.PR↗

arXiv2022

Fluctuations for mean field limits of interacting systems of spiking neurons

We consider a system of $N$ neurons, each spiking randomly with rate depending on its membrane potential. When a neuron spikes, its potential is reset to $0$ and all other neurons receive an additional amount $h/N$ of potential, where $ h > 0$ is some fixed parameter. In between successive spikes, each neuron's potential undergoes some leakage at constant rate $ α. $ While the propagation of chaos of the system, as $N \to \infty$, to a limit nonlinear jumping stochastic differential equation has already been established in a series of papers, see De Masi et al. (2015) and Fournier and Löcherbach (2016), the present paper is devoted to the associated central limit theorem. More precisely we study the measure valued process of fluctuations at scale $ N^{-1/2}$ of the empirical measures of the membrane potentials, centered around the associated limit. We show that this fluctuation process, interpreted as càdlàg process taking values in a suitable weighted Sobolev space, converges in law to a limit process characterized by a system of stochastic differential equations driven by Gaussian white noise. We complete this picture by studying the fluctuations, at scale $ N^{-1/2}, $ of a fixed number of membrane potential processes around their associated limit quantities, giving rise to a mesoscopic approximation of the membrane potentials that take into account the correlations within the finite system.

math.PR↗

arXiv2021

On decay-surge population models

We consider continuous space-time decay-surge population models which are semi- stochastic processes for which deterministically declining populations, bound to fade away, are rein- vigorated at random times by bursts or surges of random sizes. In a particular separable framework (in a sense made precise below) we provide explicit formulae for the scale (or harmonic) function and the speed measure of the process. The behavior of the scale function at infinity allows to formulate conditions under which such processes either explode or are transient at infinity, or Harris recurrent. A description of the structures of both the discrete-time embedded chain and extreme record chain of such continuous-time processes is supplied.

math.PR↗

arXiv2021

On population growth with catastrophes

In this paper we study a particular class of Piecewise deterministic Markov processes (PDMP's) which are semi-stochastic catastrophe versions of deterministic population growth models. In between successive jumps the process follows a flow describing deterministic population growth. Moreover, at random jump times, governed by state-dependent rates, the size of the population shrinks by a random amount of its current size, an event possibly leading to instantaneous local (or total) extinction. A special separable shrinkage transition kernel is investigated in more detail, including the case of total disasters. We discuss conditions under which such processes are recurrent (positive or null) or transient. To do so, we introduce a modified scale function which is used to compute, when relevant, the law of the height of excursions and to decide if the process is recurrent or not. The question of the finiteness of the time to extinction is investigated together with the evaluation of the mean time to extinction when the last one is finite. Some information on the embedded jump chain of the PDMP is also required when dealing with the classification of states 0 and infinity that we exhibit.

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

Metastability for systems of interacting neurons

We study a stochastic system of interacting neurons and its metastable properties. The system consists of $N$ neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the neuron potential is reset to $0$ and all other neurons receive an additional amount $h/N$ of potential. In between successive spike times, each neuron looses potential at exponential speed. We study this system in the supercritical regime, that is, for sufficiently high values of the synaptic weight $h.$ Under very mild conditions on the spiking rate function, is has been shown in Duarte and Ost \cite{do} that the only invariant distribution of the finite system is the trivial measure $ δ_{\bf 0}$ corresponding to extinction of the process. Under minimal conditions on the behavior of the spiking rate function in the vicinity of $0$, we prove that the extinction time arrives at exponentially late times in $ N$, and discuss the stability of the equilibrium $δ_{\bf 0}$ for the non-linear mean-field limit process depending on the parameters of the dynamics. We then specify our study to the case of saturating spiking rates and show that, under suitable conditions on the parameters of the model, 1) the non-linear mean-field limit admits a unique and globally attracting equilibrium and 2) the rescaled exit times for the mean spiking rate of a finite system from a neighbourhood of the non-linear equilibrium rate converge in law to an exponential distribution, as the system size diverges. In other words, the system exhibits a metastable behavior.

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↗