We consider an infinite lattice system of interacting spins living on a smooth compact manifold, with short- but not necessarily finite-range pairwise interactions. We construct the gradient flow of the infinite-volume free energy on the space of translation-invariant spin measures, using an adaptation of the variational approach in Wasserstein space pioneered by Jordan, Kinderlehrer, and Otto. We also construct the infinite-volume diffusion corresponding to the so-called overdamped Langevin dynamics of the spins under the effect of the interactions and of thermal agitation. We show that the trajectories of the gradient flow and of the law of the spins under this diffusion both satisfy, in a weak sense, the same hierarchy of coupled parabolic PDE’s, which we interpret as an infinite-volume Fokker-Planck-Kolmogorov equation. We prove regularity of weak solutions and derive an Evolution Variational Inequality for regular solutions, which implies uniqueness. Thus, in particular, the trajectories of the gradient flow coincide with those obtained from the Langevin dynamics. Concerning the long-time evolution, we check that the free energy is always non-increasing along the flow and that moreover, if the Ricci curvature of the spin space is uniformly positive, then at high enough temperature the dynamics converges exponentially, in free energy and in specific Wasserstein distance, to the unique minimizer of the infinite-volume free energy.

We characterize the limiting distributions of random variables of the form Pn((Xi)i≥1) , where: (i) (Pn)n≥1 is a sequence of multivariate polynomials, each potentially involving countably many variables; (ii) there exists a constant D≥1 such that for all n≥1 , the degree of Pn is bounded above by D; (iii) (Xi)i≥1 is a sequence of independent and identically distributed random variables, each with zero mean, unit variance, and finite moments of all orders. More specifically, we prove that the limiting distributions of these random variables can always be represented as the law of P∞((Xi,Gi)i≥1), where P∞ is a polynomial of degree at most D (potentially involving countably many variables), and (Gi)i≥1 is a sequence of independent standard Gaussian random variables, which is independent of (Xi)i≥1 . We solve this problem in full generality, addressing both Gaussian and non-Gaussian inputs, and with no extra assumption on the coefficients of the polynomials. In the Gaussian case, our proof builds upon several original tools of independent interest, including a new criterion for central convergence based on the concept of maximal directional influence. Beyond asymptotic normality, this novel notion also enables us to derive quantitative bounds on the degree of the polynomial representing the limiting law. We further develop techniques regarding asymptotic independence and dimensional reduction. To conclude for polynomials with non-Gaussian inputs, we combine our findings in the Gaussian case with invariance principles.

Motivated by a problem from incompressible fluid mechanics of Brenier (JAMS 1989), and its recent entropic relaxation by Arnaudo, Cruizero, Léonard & Zambrini (AIHP PS 2020), we study a problem of entropic minimization on the path space when the reference measure is a generic Feller semimartingale. We show that, under some regularity condition, our problem connects naturally with a, possibly non-local, version of the Hamilton-Jacobi-Bellman equation. Additionally, we study existence of minimizers when the reference measure in a Ornstein-Uhlenbeck process.

In this article, we prove quantitative Gaussian fluctuations for linear statistics of beta ensembles, in the single cut and non-critical regime. We establish our bounds for strong probabilistic distances, such as teh total variation distance, which, to the best of our knowledge is new ate this level of generality, even qualitatively. Provided the test functions is fussiciently smooth, we recover the sharp speed 1/n. Under the same assumptions, we also establish multivariate CLTs for vectors of linear statistics in p−Wasserstein distances for any p≥1, with the optimal rate 1/n. Second, we establish the so-called super-convergence of linear statistics, namely the convergence of all derivatives of the densities uniformly, provided that the test functions is not too flat.

We present a new tool to study the regularity of a function F of a sequence (Xi) of independent and identically distributed random variables. Our main result states that, under mild conditions on the law of X1, the regularity of the law of F is controlled by the regularity of the law of a conditionally Gaussian object, canonically associated with F. At the technical level our analysis relies on the formalism of Dirichlet forms and an explicit construction of the Malliavin derivative of F in the direction of a Gaussian space. As an application, we derive an explicit control of the regularity of the law of a quadratic from in the Xi’s in terms of spectral quantities, when the law of X1 belongs to a large class of distribution including, for instance, all the Gaussian, all the Beta, all the Gamma, and all the polynomials thereof.

We show that on Wiener chaoses, asymptotic normality always implies asymptotic smoothness of the density. In particular, this implies, that without further assumption, every sequence of Wiener chaoses converging in law to a Gaussian has automatically their densities, all well as all their derivatives, converging uniformly to those of a Gaussian.

We use optimal transport ideas to study the geometry of the space of discrete measures over a separable metric space, equipped with the Poisson measure. In particular, we construct a variational distance of this space, and establish non-local infinite dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below:

• the Ornstein–Uhlenbeck semi-group is the gradient flow of the relative entropy;
• the Poisson space has a Ricci curvature, in the entropic sense, bounded below by 1;
• the distance W satisfies an HWI inequality.

On the torus of dimension n, we study various possible discretisations of the poly-harmonic Gaussian field, that is the random Gaussian field whose covariance is the inverse of the Laplacian to the n/2. We also study the associated Gaussian multiplicative chaoses and the convergence thereof.

We prove the strong form of the Gaussian product conjecture in dimension three. Our purely analytical proof simplifies previously known proofs based on combinatorial methods or computer-assisted methods, and allows us to solve the case of any triple of even positive integers which remained open so far.

On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of k distinct sets. We show that the k-th eigenvalues of the metric Laplacian gives exponential improved concentration with k sets. On compact Riemannian manifolds, this allows us to recover estimates on the eigenvalues of the Laplace-Beltrami operator in the spirit of an inequality of Chung-Grigoryan & Yau.

We derive transport-entropy inequalities for mixed binomial point processes, and for Poisson point processes. We show that when the finite intensity measure satisfies a Talagrand transport inequality, the law of the point process also satisfies a Talagrand type transport inequality. We also show that a Poisson point process (with arbitrary σ-finite intensity measure) always satisfies a universal transport-entropy inequality à la Marton. We explore the consequences of these inequalities in terms of concentration of measure and modified logarithmic Sobolev inequalities. In particular, our results allow one to extend a deviation inequality by Reitzner, originally proved for Poisson random measures with finite mass.

We prove limit theorems for functionals of a Poisson point process using the Malliavin calculus on the Poisson space. The target distribution is conditionally either a Gaussian vector or a Poisson random variable. The convergence is stable and our conditions are expressed in terms of the Malliavin operators. For conditionally Gaussian limits, we also obtain quantitative bounds, given for the Monge-Kantorovich transport distance in the univariate case; and for another probabilistic variational distance in higher dimension. Our work generalizes several limit theorems on the Poisson space, including the seminal works by Peccati, Solé, Taqqu & Utzet for Gaussian approximations; and by Peccati for Poisson approximations; as well as the recently established fourth-moment theorem on the Poisson space of Döbler & Peccati. We give an application to stochastic processes.

On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of k distinct sets. We show that the k-th eigenvalues of the metric Laplacian gives exponential improved concentration with k sets. On compact Riemannian manifolds, this allows us to recover estimates on the eigenvalues of the Laplace-Beltrami operator in the spirit of an inequality of Chung-Grigoryan & Yau.

In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves a Brownian bridge on the Heisenberg group. This new inequality contains the optimal logarithmic Sobolev inequality for the Gaussian distribution in two dimensions. We compare this new inequality with the sub-elliptic logarithmic Sobolev inequality of Hong-Quan Li and with the more recent inequality of Fabrice Baudoin and Nicola Garofalo obtained using a generalized curvature criterion. Finally, we extend this inequality to the case of homogeneous Carnot groups of rank two.