arXiv subjects

Armand Bernou

Publications and source records attributed to Armand Bernou.

12 recordsLinked to original sources

arXiv2025

Mean-field approximation, Gibbs relaxation, and cross estimates

We study the propagation of chaos and relaxation to Gibbs equilibrium for a system of $N$ classical Brownian particles with weak mean-field interactions. It is well known that propagation of chaos holds uniformly in time with rate $O(N^{-1})$ and that Gibbs relaxation holds uniformly in $N$ with exponential rate $O(e^{-ct})$. We go one step further by establishing a cross estimate that simultaneously captures both effects: the joint deviation between chaos propagation and Gibbs relaxation is of order $O(N^{-1}e^{-ct})$. In particular, for translation-invariant systems, this yields an accelerated propagation of chaos, with the mean-field approximation error at the level of the one-particle density improving from $O(N^{-1})$ to $O(N^{-1}e^{-ct})$. Our approach relies on a detailed analysis of the BBGKY hierarchy for correlation functions, and applies to both underdamped and overdamped Langevin dynamics with merely bounded interaction forces. In addition, we obtain new quantitative results on Gibbs relaxation and provide partial extensions beyond the weak interaction regime.

math.AP↗

arXiv2025

Creation of chaos for interacting Brownian particles

We consider a system of $N$ Brownian particles, with or without inertia, interacting in the mean-field regime via a weak, smooth, long-range potential, and starting initially from an arbitrary exchangeable $N$-particle distribution. In this model framework, we establish a fine version of the so-called creation-of-chaos phenomenon: in weak norms, the mean-field approximation for a typical particle is shown to hold with an accuracy $O(N^{-1})$ up to an error due solely to initial pair correlations, which is damped exponentially over time. Corresponding higher-order results are also derived in the form of higher-order correlation estimates. The approach is new and easily adaptable: we start from suboptimal correlation estimates obtained from an elementary use of Itô's calculus on moments of the empirical measure, together with ergodic properties of the mean-field dynamics, and these bounds are then made optimal after combination with PDE estimates on the BBKY hierarchy.

math.PR↗

arXiv2025

Convex order and increasing convex order for McKean-Vlasov processes with common noise

We establish results on the conditional and standard convex order, as well as the increasing convex order, for two processes $ X = (X_t)_{t \in [0, T]} $ and $ Y = (Y_t)_{t \in [0, T]} $, defined by the following McKean-Vlasov equations with common Brownian noise $ B^0 = (B_t^0)_{t \in [0, T]} $: $$ dX_t=b(t, X_t, \mathcal{L}^1(X_t))d t+\sigma(t, X_t, \mathcal{L}^1(X_t))d B_t+\sigma^0 (t, \mathcal{L}^1(X_t))d B^0_t$$ $$dY_t=\,\beta(t, Y_t, \mathcal{L}^1(Y_t\,))d t+\,\theta(t, Y_t\,, \mathcal{L}^1(Y_t\,))d B_t\,+\,\theta^0 (t, \mathcal{L}^1(Y_t\,))d B^0_t,$$ where $ \mathcal{L}^1(X_t) $ (respectively $ \mathcal{L}^1(Y_t) $) denotes a version of the conditional distribution of $ X_t $ (resp. $ Y_t $) given $ B^0 $. These results extend those established for standard McKean-Vlasov equations in [Liu-Pag\`es, 2023] and [Liu-Pag\`es, 2021]. Under suitable conditions, for a (non-decreasing) convex functional $F$ on the path space with polynomial growth, we show $ \mathbb{E}[F(X) | B^0] \leq \mathbb{E}[F(Y) | B^0] $ almost surely. Moreover, for a (non-decreasing) convex functional $G$ defined on the product space of paths and their marginal distributions, we establish $$ \mathbb{E} \Big[\,G\big(X, (\mathcal{L}^1(X_t))_{t\in[0, T]}\big)\,\Big| \, B^0\,\Big]\leq \mathbb{E} \Big[\,G\big(Y, (\mathcal{L}^1(Y_t))_{t\in[0, T]}\big)\,\Big| \, B^0\,\Big] \quad \text{almost surely}. $$ Similar convex order results are also established for the corresponding particle system. Finally, we explore applications of these results to stochastic control problems and to the interbank systemic risk model introduced in [Carmona-Fouque-Sun, 2015].

math.PR↗

arXiv2025

Uniform-in-time estimates on corrections to mean field for interacting Brownian particles

We consider a system of classical Brownian particles interacting via a smooth long-range potential in the mean-field regime, and we analyze the propagation of chaos in form of sharp, uniform-in-time estimates on many-particle correlation functions. Our results cover both the kinetic Langevin setting and the corresponding overdamped Brownian dynamics. The approach is mainly based on so-called Lions expansions, which we combine with new diagrammatic tools to capture many-particle cancellations, as well as with fine ergodic estimates on the linearized mean-field equation, and with discrete stochastic calculus with respect to initial data. In the process, we derive some new ergodic estimates for the linearized Vlasov-Fokker-Planck kinetic equation that are of independent interest. Our analysis also leads to a uniform-in-time quantitative central limit theorem and to uniform-in-time concentration estimates for the empirical measure associated with the particle dynamics.

math.AP↗

arXiv2024

Asymptotic Behavior of Degenerate Linear Kinetic Equations with Non-Isothermal Boundary Conditions

We study the degenerate linear Boltzmann equation inside a bounded domain with a generalized diffuse reflection at the boundary and variable temperature, including the Maxwell boundary conditions with the wall Maxwellian or heavy-tailed reflection kernel and the Cercignani-Lampis boundary condition. Our abstract collisional setting applies to the linear BGK model, the relaxation towards a space-dependent steady state, and collision kernels with fat tails. We prove for the first time the existence of a steady state and a rate of convergence towards it without assumptions on the temperature variations. Our results for the Cercignani-Lampis boundary condition make also no hypotheses on the accommodation coefficients. The proven rate is exponential when a control condition on the degeneracy of the collision operator is satisfied, and only polynomial when this assumption is not met, in line with our previous results regarding the free-transport equation. We also provide a precise description of the different convergence rates, including lower bounds, when the steady state is bounded. Our method yields constructive constants.

math.AP↗

arXiv2023

Homogenization of active suspensions and reduction of effective viscosity

We consider a suspension of active rigid particles (swimmers) in a steady Stokes flow, where particles are distributed according to a stationary ergodic random process, and we study its homogenization in the macroscopic limit. A key point in the model is that swimmers are allowed to adapt their propulsion to the surrounding fluid deformation: swimming forces are not prescribed a priori, but are rather obtained through the retroaction of the fluid. Qualitative homogenization of this nonlinear model requires an unusual proof that crucially relies on a semi-quantitative two-scale analysis. After introducing new correctors that accurately capture spatial oscillations created by swimming forces, we identify the contribution of the activity to the effective viscosity. In agreement with the physics literature, an analysis in the dilute regime shows that the activity of the particles can either increase or decrease the effective viscosity (depending on the swimming mechanism), which differs from the well-known case of passive suspensions.

math.AP↗

arXiv2022

Particle method for the numerical simulation of the path-dependent McKean-Vlasov equation

We present the particle method for simulating the solution to the path-dependent McKean-Vlasov equation, in which both the drift and the diffusion coefficients depend on the whole trajectory of the process up to the current time t, as well as on the corresponding marginal distributions. Our main contribution is the derivation of explicit convergence rates that capture the interplay between time and space discretization. To control the uniform-in-time convergence of empirical measures in Wasserstein distance on a fixed interval, we develop two approaches: one based on Fournier-Guillin estimates, the other extending Horowitz-Karandikar's method to general p greater or equal to 2. We then compare the respective regimes of applicability. Numerical simulations of a generalized Ornstein-Uhlenbeck process with memory provide evidence for the accuracy of our bounds. We also apply our method to an extension of the Jansen-Rit mean-field model for neural masses.

math.PR↗

arXiv2021

On Subexponential Convergence to Equilibrium of Markov Processes

Studying the subexponential convergence towards equilibrium of a strong Markov process, we exhibit an intermediate Lyapunov condition equivalent to the control of some moment of a hitting time. This provides a link, similar (although more intricate) to the one existing in the exponential case, between the coupling method and the approach based on the existence of a Lyapunov function for the generator, in the context of the subexponential rates found by Fort-Roberts (2005), Douc-Fort-Guillin (2009) and Hairer (2016).

math.PR↗

arXiv2021

Convergence Towards the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition

We study the asymptotic behavior of the kinetic free-transport equation enclosed in a regular domain, on which no symmetry assumption is made, with Cercignani-Lampis boundary condition. We give the first proof of existence of a steady state in the case where the temperature at the wall varies, and derive the optimal rate of convergence towards it, in the L1 norm. The strategy is an application of a deterministic version of Harris subgeometric theorem, in the spirit of Cañizo-Mischler (2021) and Bernou (2020). We also investigate rigorously the velocity flow of a model mixing pure diffuse and Cercignani-Lampis boundary conditions with variable temperature, for which we derive an explicit form for the steady state, providing new insights on the role of the Cercignani-Lampis boundary condition in this problem.

math.AP↗

arXiv2021

Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition

We establish the convergence to the equilibrium for various linear collisional kinetic equations (including linearized Boltzmann and Landau equations) with physical local conservation laws in bounded domains with general Maxwell boundary condition. Our proof consists in establishing an hypocoercivity result for the associated operator, in other words, we exhibit a convenient Hilbert norm for which the associated operator is coercive in the orthogonal of the global conservation laws. Our approach allows us to treat general domains with all type of boundary conditions in a unified framework. In particular, our result includes the case of vanishing accommodation coefficient and thus the specific case of the specular reflection boundary condition.

math.AP↗

arXiv2020

A semigroup approach to the convergence rate of a collisionless gas

We study the rate of convergence to equilibrium for a collisionless (Knudsen)gas enclosed in a vessel in dimension $n \in \{2,3\}$. By semigroup arguments,we prove that in the $L^1$ norm, the polynomial rate of convergence$\frac{1}{(t+1)^{n-}}$ given by Tsuji et al [2010] and Kuo et al[2013,2014,2015] can be extended to any $C^2$ domain, with standard assumptionson the initial data. This is to our knowledge, the first quantitative result incollisionless kinetic theory in dimension equal to or larger than 2 relying ondeterministic arguments that does not require any symmetry of the domain, nor amonokinetic regime. The dependency of the rate with respect to the initialdistribution is detailed. Our study includes the case where the temperature atthe boundary varies. The demonstrations are adapted from a deterministicversion of a subgeometric Harris' theorem recently established by Cañizo andMischler. We also compare our model with a free-transport equation withabsorbing boundary.

math.AP↗

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↗