Central limit theorem over non-linear functionals of empirical measures with applications to the mean-field fluctuation of interacting particle systems
In this work, a generalised version of the central limit theorem is proposed for nonlinear functionals of the empirical measure of i.i.d. random variables, provided that the functional satisfies some regularity assumptions for the associated linear functional derivative. This generalisation can be applied to Monte-Carlo methods, even when there is a nonlinear dependence on the measure component. We use this result to deal with the contribution of the initialisation in the convergence of the fluctuations between the empirical measure of interacting diffusion and their mean-field limiting measure (as the number of particles goes to infinity), when the dependence on measure is nonlinear. A complementary contribution related to the time evolution is treated using the master equation, a parabolic PDE involving L-derivatives with respect to the measure component, which is a stronger notion of derivative that is nonetheless related to the linear functional derivative.
We prove density and current fluctuations for two examples of symmetric, interacting particle systems with anomalous diffusive behavior: the zero-range process with long jumps and the zero-range process with degenerated bond disorder. As an application, we obtain subdiffusive behavior of a tagged particle in a simple exclusion process with variable diffusion coefficient.
We study equilibrium fluctuations for a class of totally asymmetric zero-range type interacting particle systems. As a main result, we show that density fluctuation of our process converges to the stationary energy solution of the stochastic Burgers equation. As a special case, microscopic system we consider here is related to $q$-totally asymmetric simple exclusion processes ($q$-TASEPs) and our scaling limit corresponds to letting the quantum parameter $q$ to be one.
Motivated by the recent preprint [arXiv:2004.08412] by Ayala, Carinci, and Redig, we first provide a general framework for the study of scaling limits of higher order fields. Then, by considering the same class of infinite interacting particle systems as in [arXiv:2004.08412], namely symmetric simple exclusion and inclusion processes in the d-dimensional Euclidean lattice, we prove the hydrodynamic limit, and convergence for the equilibrium fluctuations, of higher order fields. In particular, the limit fields exhibit a tensor structure. Our fluctuation result differs from that in [arXiv:2004.0…
The propagation of chaos and associated law of large numbers for mean-field interacting age-dependent Hawkes processes (when the number of processes n goes to +$\infty$) being granted by the study performed in (Chevallier, 2015), the aim of the present paper is to prove the resulting functional central limit theorem. It involves the study of a measure-valued process describing the fluctuations (at scale n --1/2) of the empirical measure of the ages around its limit value. This fluctuation process is proved to converge towards a limit process characterized by a limit system of stochastic differ…
We investigate the asymptotic behaviour of networks of interacting non-linear Hawkes processes modeling a homogeneous population of neurons in the large population limit. In particular, we prove a functional central limit theorem for the mean spike-activity thereby characterizing the asymptotic fluctuations in terms of a stochastic Volterra integral equation. Our approach differs from previous approaches in making use of the associated resolvent in order to represent the fluctutations as Skorokhod continuous mappings of weakly converging martingales. Since the Lipschitz properties of the resol…
We consider large fluctuations, namely overload of servers, in a network with dynamic routing of messages. The servers form a circle. The number of input flows is equal to the number of servers, the messages of any flow are distributed between two neighboring servers, upon its arrival a message is directed to the least loaded of these servers. Under the condition that at least two servers are overloaded the number of overloaded servers in such network depends on the rate of input flows. In particular there exists critical level of input rate that in case of higher rate most probable that all s…
We consider a model of N two-colors urns in which the reinforcement of each urn depends also on the content of all the other urns. This interaction is of mean-field type and it is tuned by a parameter $α$ in [0,1]; in particular, for $α=0$ the N urns behave as N independent Polya's urns. As shown in [9], for $α>0$ urns synchronize, in the sense that the fraction of balls of a given color converges a.s. to the same (random) limit in all urns. In this paper we study fluctuations around this synchronized regime. The scaling of these fluctuations depends on the parameter $α$. In particular the sta…
We establish uniqueness for a class of first-order Hamilton-Jacobi equations with Hamiltonians that arise from the large deviations of the empirical measure and empirical flux pair of weakly interacting Markov jump processes. As a corollary we obtain a large deviation principle for the trajectory of the empirical measure and empirical flux pair of such processes. As a second corollary we get same result in the setting where the jump-rates are time-periodic, with period-length that decreases to 0.
We investigate the fluctuations around the mean of the Stieltjes transform of the empirical spectral distribution of any selfadjoint noncommutative polynomial in a Wigner matrix and a deterministic diagonal matrix. We obtain the convergence in distribution to a centred complex Gaussian process whose covariance is expressed in terms of operator-valued subordination functions.
We introduce what we call the second-order Boltzmann-Gibbs principle, which allows to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear function of the conserved quantity. This replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states. As an application of this second-order Boltzmann-Gibbs principle, we introduce the notion of energy solutions of the KPZ and stochastic Burgers equations. Under minimal assumptions, we prove that the density fluctuat…
A collection of $N$-diffusing interacting particles where each particle belongs to one of $K$ different populations is considered. Evolution equation for a particle from population $k$ depends on the $K$ empirical measures of particle states corresponding to the various populations and the form of this dependence may change from one population to another. In addition, the drift coefficients in the particle evolution equations may depend on a factor that is common to all particles and which is described through the solution of a stochastic differential equation coupled, through the empirical me…
Consider a collection of particles whose state evolution is described through a system of interacting diffusions in which each particle is driven by an independent individual source of noise and also by a small amount of noise that is common to all particles. The interaction between the particles is due to the common noise and also through the drift and diffusion coefficients that depend on the state empirical measure. We study large deviation behavior of the empirical measure process which is governed by two types of scaling, one corresponding to mean field asymptotics and the other to the Fr…
A basic result of large deviations theory is Sanov's theorem, which states that the sequence of empirical measures of independent and identically distributed samples satisfies the large deviation principle with rate function given by relative entropy with respect to the common distribution. Large deviation principles for the empirical measures are also known to hold for broad classes of weakly interacting systems. When the interaction through the empirical measure corresponds to an absolutely continuous change of measure, the rate function can be expressed as relative entropy of a distribution…
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 Löcherba…
We investigate in this work the effects of interaction on the fluctuation of empirical measures. The systems with positive definite interaction potentials tend to exhibit smaller fluctuation compared to the fluctuation in standard Monte Carlo sampling while systems with negative definite potentials tend to exhibit larger fluctuation. Moreover, if the temperature goes to zero, the fluctuation for positive definite kernels in the long time tends to vanish to zero, while the fluctuation for negative definite kernels in the long time tends to blow up to infinity. This phenomenon may gain deeper un…
We compute the Hamiltonian and Lagrangian associated to the large deviations of the trajectory of the empirical distribution for independent Markov processes, and of the empirical measure for translation invariant interacting Markov processes. We treat both the case of jump processes (continuous-time Markov chains and interacting particle systems) as well as diffusion processes. For diffusion processes, the Lagrangian is a quadratic form of the deviation of the trajectory from the Kolmogorov forward equation. In all cases, the Lagrangian can be interpreted as a relative entropy or relative ent…
The Kuramoto model has been introduced to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc. The model consists of $N$ interacting oscillators on the one dimensional sphere $\mathbf{S}^{1}$, driven by independent Brownian Motions with constant drift chosen at random. This quenched disorder is chosen independently for each oscillator according to the same law $μ$. The behaviour of the system for large $N$ can be understood via its empirical measure: we prove here the convergence as $N\to\infty$ of the quenched empirical measure to the unique solution of …
In this work, we consider a finite-state inhomogeneous-time Markov chain whose probabilities of transition from one state to another tend to decrease over time. This can be seen as a cooling of the dynamics of an underlying Markov chain. We are interested in the long time behavior of the empirical measure of this freezing Markov chain. Some recent papers provide almost sure convergence and convergence in distribution in the case of the freezing speed $n^{-θ}$, with different limits depending on $θ<1,θ=1$ or $θ>1$. Using stochastic approximation techniques, we generalize these results for any f…
In this paper, we consider isotropic and stationary real Gaussian random fields defined on $\mathbb{S}^2\times\mathbb{R}$ and we investigate the asymptotic behavior, as $T\rightarrow +\infty$, of the empirical measure (excursion area) in $\mathbb{S}^2\times\mathbb{R}$ at any threshold, covering both cases when the field exhibits short and long memory, i.e. integrable and non-integrable temporal covariance. It turns out that the limiting distribution is not universal, depending both on the memory parameters and the threshold. In particular, in the long memory case a form of Berry's cancellation…
We present a multivariate central limit theorem for a general class of interacting Markov chain Monte Carlo algorithms used to solve nonlinear measure-valued equations. These algorithms generate stochastic processes which belong to the class of nonlinear Markov chains interacting with their empirical occupation measures. We develop an original theoretical analysis based on resolvent operators and semigroup techniques to analyze the fluctuations of their occupation measures around their limiting values.
We study central limit theorems for a totally asymmetric, one-dimensional interacting random system. The models we work with are the Aldous-Diaconis-Hammersley process and the related stick model. The A-D-H process represents a particle configuration on the line, or a 1-dimensional interface on the plane which moves in one fixed direction through random local jumps. The stick model is the process of local slopes of the A-D-H process, and has a conserved quantity. The results describe the fluctuations of these systems around the deterministic evolution to which the random system converges under…
We obtain the large scale limit of the fluctuations around its hydrodynamic limit of the density of particles of a weakly asymmetric exclusion process in dimension up to three. The proof is based upon a sharp estimate on the relative entropy of the law of the process with respect to product reference measures associated to the hydrodynamic limit profile, which holds in any dimension and is of independent interest. As a corollary of this entropy estimate, we obtain some quantitative bounds on the speed of convergence of the aforementioned hydrodynamic limit.
We study fluctuations of the empirical processes of a non-equilibrium interacting particle system consisting of two species over a domain that is recently introduced in [8] and establish its functional central limit theorem. This fluctuation limit is a distribution-valued Gaussian Markov process which can be represented as a mild solution of a stochastic partial differential equation. The drift of our fluctuation limit involves a new partial differential equation with nonlinear coupled term on the interface that characterized the hydrodynamic limit of the system. The covariance structure of th…
This is the second of two articles on the study of a particle system model that exhibits a Turing instability type effect. About the hydrodynamic quations obtained in \cite{CSL17a}, we find conditions under which Turing instability occurs around the null equilibrium solution. In this instability regime: for long times at which the process is of infinitesimal order, we prove that the non-equilibrium fluctuations around the hydrodynamic limit are Gaussian; for times converging to the critical one at which the process is of finite order, we prove that the $\pm 1$-Fourier modes are uniformly away …
This paper addresses the question of the fluctuations of the empirical entropy of a chain of infinite order. We assume that the chain takes values on a finite alphabet and loses memory exponentially fast. We consider two possible definitions for the empirical entropy, both based on the empirical distribution of cylinders with length clog(n), where n is the size of the sample and c is a suitable constant. The first one is the conditional entropy of the empirical distribution, given a past with length growing logarithmically with the size of the sample. The second one is the rescaled entropy of …
We consider systems of $N$ diffusions in equilibrium interacting through a potential $V$. We study a "height function" which for the special choice $V(x) = \e^{-x}$, coincides with the partition function of a stationary semidiscrete polymer, also known as the (stationary) O'Connell-Yor polymer. For a general class of smooth convex potentials (generalizing the O'Connell-Yor case), we obtain the order of fluctuations of the height function by proving matching upper and lower bounds for the variance of order $N^{2/3}$, the expected scaling for models lying in the KPZ universality class. The model…
We consider a financial market model which consists of a financial asset and a large number of interacting agents classified into many types. Different types of agents are heterogeneous in their price expectations. Each agent can change its type based on the current empirical distribution of the types and the equilibrium price, and the equilibrium price follows a recursive price mechanism based on the previous price and the current empirical distribution of the types. The interaction among the agents, and the interaction between the agents and the equilibrium price, feedback, are modeled. We a…
We study current fluctuations in lattice gases in the hydrodynamic scaling limit. More precisely, we prove a large deviation principle for the empirical current in the symmetric simple exclusion process with rate functional I. We then estimate the asymptotic probability of a fluctuation of the average current over a large time interval and show that the corresponding rate function can be obtained by solving a variational problem for the functional I. For the symmetric simple exclusion process the minimizer is time independent so that this variational problem can be reduced to a time independen…
Conjecture II.3.6 of Spohn in [Spohn '91] and Lecture 7 of Jensen-Yau in [Jensen-Yau '99] ask for a general derivation of universal fluctuations of hydrodynamic limits in large-scale stochastic interacting particle systems. However, the past few decades have witnessed only minimal progress according to [Goncalves-Landim-Milanes '17]. In this paper, we develop a general method for deriving the so-called Boltzmann-Gibbs principle for a general family of non-integrable and non-stationary interacting particle systems, thereby responding to Spohn and Jensen-Yau. Most importantly, our method depends…
For non-equilibrium systems of interacting particles and for interacting diffusions in d dimensions, a novel fluctuation relation is derived. The theorem establishes a quantitative relation between the probabilities of observing two current values in different spatial directions. The result is a consequence of spatial symmetries of the microscopic dynamics, generalizing in this way the Gallavotti-Cohen fluctuation theorem related to the time-reversal symmetry. This new perspective opens up the possibility of direct experimental measurements of fluctuation relations of vectorial observables.
The aim of the paper is to establish a large deviation principle (LDP) for the empirical measure of mean-field interacting diffusions in a random environment. The point is to derive such a result once the environment has been frozen (quenched model). The main theorem states that a LDP holds for every realization of the environment, with a rate function that does not depend on the disorder and is different from the rate function in the averaged model. Similar results concerning the empirical flow and local empirical measures are provided.
We consider a non-integrable model for interacting dimers on the two-dimensional square lattice. Configurations are perfect matchings of $\mathbb Z^2$, i.e. subsets of edges such that each vertex is covered exactly once ("close-packing" condition). Dimer configurations are in bijection with discrete height functions, defined on faces $\boldsymbolξ$ of $\mathbb Z^2$. The non-interacting model is "integrable" and solvable via Kasteleyn theory; it is known that all the moments of the height difference $h_{\boldsymbolξ}-h_{\boldsymbolη}$ converge to those of the massless Gaussian Free Field (GFF),…
In this short note we consider semi-Markov processes satisfying the condition of direction-time independence (Markov renewal processes). We derive large deviation principles and fluctuation theorems for the empirical current and the empirical currents along cycles. Our derivation is based on the joint LDP for the empirical measure and flow recently proved in \cite{MZ}.
