This paper is concerned with the mean-field limit for the gradient flow evolution of particle systems with pairwise Riesz interactions, as the number of particles tends to infinity. Based on a modulated energy method, using regularity and stability properties of the limiting equation, as inspired by the work of Serfaty [28] in the context of the Ginzburg-Landau vortices, we prove a mean-field limit result in dimensions 1 and 2 in cases for which this problem was still open. IntroductionWe consider the energy of a system of N particles in the Euclidean space R d (d ≥ 1) interacting via (repulsive) Riesz pairwise interactions:where the interaction kernel is given bywith c d,s > 0 some normalization constants. We note that the Coulomb case corresponds to the choice s = d − 2, d ≥ 2. Particle systems with more general Riesz interactions as considered here are extensively motivated in the physics literature (cf. for instance [2,20]), as well as in the context of approximation theory with the study of Fekete points (cf.[14] and the references therein). Recently, a detailed description of such systems beyond the mean-field limit in the static case was obtained in [21], and also in [17] for non-zero temperature. In the present contribution, we are rather interested in the dynamics of such systems, and more precisely in a rigorous justification of the mean-field limit of their gradient flow evolution as the number N of particles tends to infinity, which has indeed remained an open problem wheneverWe thus consider the trajectories x t i,N driven by the corresponding flow, i.e. the solutions to the following system of ODEs:where (xis a sequence of N distinct initial positions. Since energy can only decrease in time and since the interaction is repulsive, particles cannot collide, and moreover it is easily seen that a particle cannot escape to infinity in finite time; from these observations and from the Picard-Lindelöf theorem, we may conclude that the trajectories x t i,N are smooth and well-defined on the whole of R + := [0, ∞). As the number of particles gets large, we would naturally like to pass to a continuum description of the system, in terms of the particle density distribution. For that purpose, we define the empirical measure associated with the point-vortex dynamics:2) 1 and the question is then to understand the limit of µ t N as N ↑ ∞. More precisely, assuming convergence at initial time This equation in the weak sense just means the following:As far as existence issues as well as basic properties of the solutions of (1.3) are concerned, we refer to [9,8] . Schochet's original paper [27] was actually only concerned with the mean-field limit for a particle approximation of the 2D Euler equation, but the same argument directly applies to the present setting. However, due to a possible lack of uniqueness of L 1 weak solutions to equation (1.3), Schochet [27] could only prove that the empirical measure µ t N converges up to a subsequence to some solution of (1.3). The key idea, which only holds for logarithmic i...

We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order d 2 , where d is the spatial dimension; the fluctuations turn out to be Gaussian. As for previous work on the leading order, this higher-order characterization relies on a pathwise proximity of the macroscopic fluctuations of a general solution to those of the (higher-order) correctors, via a (higher-order) two-scale expansion injected into the "homogenization commutator", thus confirming the scope of this notion. This higher-order generalization sheds a clearer light on the algebraic structure of the higher-order versions of correctors, flux correctors, two-scale expansions, and homogenization commutators. It reveals that in the same way as this algebra provides a higher-order theory for microscopic spatial oscillations, it also provides a higher-order theory for macroscopic random fluctuations, although both phenomena are not directly related. We focus on the model framework of an underlying Gaussian ensemble, which allows for an efficient use of (second-order) Malliavin calculus for stochastic estimates. On the technical side, we introduce annealed Calderón-Zygmund estimates for the elliptic operator with random coefficients, which conveniently upgrade the known quenched large-scale estimates.

Four quantities are fundamental in homogenization of elliptic systems in divergence form and in its applications: the field and the flux of the solution operator (applied to a general deterministic right-hand side), and the field and the flux of the corrector. Homogenization is the study of the large-scale properties of these objects. For random coefficients, these quantities fluctuate and their fluctuations are a priori unrelated. Depending on the law of the coefficient field, and in particular on the decay of its correlations on large scales, these fluctuations may display different scalings and different limiting laws (if any). In this contribution, we identify a fifth and crucial intrinsic quantity, a random 2-tensor field, which we refer to as the homogenization commutator. In the model framework of discrete linear elliptic equations in divergence form with independent and identically distributed coefficients, we show what we believe to be a general principle, namely that the homogenization commutator drives at leading order the fluctuations of each of the four other quantities in probability, which reveals the pathwise structure of fluctuations in stochastic homogenization. In addition, we show in this framework that the (rescaled) homogenization commutator converges in law to a (2-tensor) Gaussian white noise, the distribution of which is thus characterized by some 4-tensor, and we analyze to which precision this tensor can be extracted from the representative volume element method. All these results are optimally quantified and hold in any dimension. This constitutes the first complete theory of fluctuations in stochastic homogenization. Extensions to the (non-symmetric) continuum setting are discussed, while details are postponed to forthcoming work.