arXiv subjects

Laetitia Colombani

Publications and source records attributed to Laetitia Colombani.

4 recordsLinked to original sources

arXiv2024

Analytic and Monte Carlo Approximations to the Distribution of the First Passage Time of the Drifted Diffusion with Stochastic Resetting and Mixed Boundary Conditions

This article introduces two techniques for computing the distribution of the absorption or first passage time of the drifted Wiener diffusion subject to Poisson resetting times, to an upper hard wall barrier and to a lower absorbing barrier. The first method, which we call ``Padé-partial fraction'' approximation, starts with the Padé approximation to the Laplace transform of the first passage time distribution, which is then exactly inverted by means of the partial fraction decomposition. The second method, which we call "multiresolution algorithm", is a Monte Carlo technique that exploits the properties of the Wiener process in order to generate Brownian bridges at increasing levels of resolution. Our numerical study reveals that the multiresolution algorithm has higher efficiency than standard Monte Carlo, whereas the faster Padé-partial fraction method is accurate in various circumstances and provides an analytical formula. Also, a closed-form exact expression for the expected first passage time is derived.

physics.comp-ph↗

arXiv2023

Propagation of chaos in mean field networks of FitzHugh-Nagumo neurons

In this article, we are interested in the behavior of a fully connected network of $N$ neurons, where $N$ tends to infinity. We assume that the neurons follow the stochastic FitzHugh-Nagumo model, whose specificity is the non-linearity with a cubic term. We prove a result of uniform in time propagation of chaos of this model in a mean-field framework. We also exhibit explicit bounds. We use a coupling method initially suggested by A. Eberle (arXiv:1305.1233), and recently extended in (1805.11387), known as the reflection coupling. We simultaneously construct a solution of the $N$-particle system and $N$ independent copies of the non-linear McKean-Vlasov limit in such a way that, considering an appropriate semi-metric that takes into account the various possible behaviors of the processes, the two solutions tend to get closer together as $N$ increases, uniformly in time. The reflection coupling allows us to deal with the non-convexity of the underlying potential in the dynamics of the quantities defining our network, and show independence at the limit for the system in mean field interaction with sufficiently small Lipschitz continuous interactions.

math.PR↗

arXiv2023

Asymptotic deviation bounds for cumulative processes

The aim of this paper is to get asymptotic deviation bounds via a Large Deviation Principle (LDP) for cumulative processes also known as compound renewal processes or renewal-reward processes. These processes cumulate independent random variables occurring in time interval given by a renewal process. Our result extends the one obtained in Lefevere et al. (2011) in the sense that we impose no specific dependency between the cumulated random variables and the renewal process and the proof uses Mariani et al. (2014). In the companion paper Cattiaux-Costa-Colombani (2021) we apply this principle to Hawkes processes with inhibition. Under some assumptions Hawkes processes are indeed cumulative processes, but they do not enter the framework of Lefevere et al. (2011).

math.PR↗

arXiv2021

Limit theorems for Hawkes processes including inhibition

In this paper we consider some non linear Hawkes processes with signed reproduction function (or memory kernel) thus exhibiting both self-excitation and inhibition. We provide a Law of Large Numbers, a Central Limit Theorem and large deviation results, as time growths to infinity. The proofs lie on a renewal structure for these processes introduced in Costa et al. (2020) which leads to a comparison with cumulative processes. Explicit computations are made on some examples. Similar results have been obtained in the literature for self-exciting Hawkes processes only.

math.PR↗