Research papers

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.