arXiv subjects

Solesne Bourguin

Publications and source records attributed to Solesne Bourguin.

At least 19 recordsLinked to original sources

arXiv2026

Densities for scalar-valued BSDEs via unique continuation and backward uniqueness

We give sufficient conditions ensuring that, at every fixed positive time, the scalar backward component of a Markovian forward-backward stochastic differential equation with multidimensional forward process admits a density with respect to Lebesgue measure. Existing density criteria for BSDEs often obtain Malliavin non-degeneracy through sign or monotonicity assumptions. We develop a different route for a scalar backward component with an arbitrary-dimensional forward state. Under regularity assumptions and a structural compatibility condition on the generator, the terminal condition is only required to be non-constant. The key idea is to deduce Malliavin non-degeneracy from deterministic rigidity of the critical set of the decoupling field. We combine Malliavin calculus with unique continuation and backward uniqueness for the associated semilinear parabolic equation. Unique continuation precludes the spatial gradient of the decoupling field from vanishing on a set of positive measure unless it vanishes identically on that time slice, while backward uniqueness propagates such vanishing to the terminal time. Along the way, we establish a unique continuation property from sets of positive measure and a backward uniqueness result on the whole space for the linear parabolic systems arising from differentiated semilinear equations.

math.PR↗

arXiv2026

Quantitative Fluctuation Analysis for Continuous-Time Stochastic Gradient Descent via Malliavin Calculus

In this paper, we establish a Quantitative Central Limit Theorem (QCLT) for the Stochastic Gradient Descent in Continuous Time (SGDCT) algorithm, whose parameter updates are governed by a stochastic differential equation. We derive an explicit rate at which the SGDCT iterates converge, in the Wasserstein metric, to a critical point of the objective function. This rate is driven primarily by the magnitude of the learning rate: for a fixed convexity constant of the objective function, smaller learning rates lead to slower convergence. Our approach relies on tools from Malliavin calculus. In particular, we apply a second-order Poincaré inequality and obtain explicit bounds by estimating the first- and second-order Malliavin derivatives separately. Controlling the second-order derivative requires several delicate calculations and a careful sequence of decompositions in order to achieve sharp estimates. We complement the theoretical results with several numerical experiments that illustrate the predicted convergence behavior.

math.PR↗

arXiv2026

Uniform-in-time quantitative fluctuations of large scale interacting particle systems

We study fluctuations of mean-field interacting particle systems around their McKean--Vlasov limit. Our main result provides a uniform-in-time quantitative central limit theorem for the fluctuation process, with convergence rate of order $N^{-1/2}$ to the corresponding Gaussian limit in the Wasserstein metric. The proof relies on two main ingredients. First, we establish a uniform-in-time weak expansion for specific functionals of the empirical measure around their limiting behavior. This yields, in particular, uniform-in-time control of the convergence of the prelimit variance to its limiting counterpart. We also derive a backward PDE representation of the limiting variance, which is of independent interest. Second, we use Malliavin calculus tools and, in particular, a second-order Poincaré inequality that bounds the Wasserstein distance between the fluctuation process and its Gaussian limit in terms of the first- and second-order Malliavin derivatives of the particle flow. The quantitative convergence rates then follow from a delicate analysis of these derivatives, yielding the sharp estimates required for uniform-in-time control.

math.PR↗

arXiv2026

Gaussian approximation on the Skorokhod space via Malliavin calculus and regularization

We introduce a carré du champ operator for Banach-valued random elements, taking values in the projective tensor product, and use it to control the bounded Lipschitz distance between a Malliavin-smooth random element satisfying mild regularity assumptions and a Radon Gaussian taking values in the Skorokhod space equipped with the uniform topology. In the case where the random element is a Banach-valued multiple integral, the carré du champ expression is further bounded by norms of the contracted integral kernel. The main technical tool is an integration by parts formula, which might be of independent interest. As a by-product, we recover a bound obtained recently by Düker and Zoubouloglou in the Hilbert space setting and complement it by providing contraction bounds.

math.PR↗

arXiv2026

A Caveat on Metrizing Convergence in Distribution on Hilbert Spaces

We consider Sobolev-type distances on probability measures over separable Hilbert spaces involving the Schatten-$p$ norms, which include as special cases a distance first introduced by Bourguin and Campese (2020) when $p=2$, and a distance introduced by Giné and Leon (1980) when $p=\infty$. Our analysis shows that, unless $p=\infty$, these distances fail to metrize convergence in distribution in infinite dimensions. This clarifies several inconsistencies and misconceptions in the recent literature that arose from confusion between different types of distances.

math.PR↗

arXiv2025

Non-central limit of densities of some functionals of Gaussian processes

We establish the convergence of the densities of a sequence of nonlinear functionals of an underlying Gaussian process to the density of a Gamma distribution. The key idea of our work is a new density formula for random variables in the setting of Markov diffusion generators, which yields a special representation for the density of a Gamma distribution. Via this representation, we are able to provide precise estimates on the distance between densities while developing the techniques of Malliavin calculus and Stein's method suitable to Gamma approximation at the density level. We first focus our study on the case of random variables living in a fixed Wiener chaos of an even order for which the bound for the difference of the densities can be dominated by a linear combination of moments up to order four. We then study the case of general Gaussian functionals with possibly infinite chaos expansion. Finally, we provide an application to random variables living in the second Wiener chaos.

math.PR↗

arXiv2025

Quantitative Error Bounds for Scaling Limits of Stochastic Iterative Algorithms

Stochastic iterative algorithms, including stochastic gradient descent (SGD) and stochastic gradient Langevin dynamics (SGLD), are widely utilized for optimization and sampling in large-scale and high-dimensional problems in machine learning, statistics, and engineering. Numerous works have bounded the parameter error in, and characterized the uncertainty of, these approximations. One common approach has been to use scaling limit analyses to relate the distribution of algorithm sample paths to a continuous-time stochastic process approximation, particularly in asymptotic setups. Focusing on the univariate setting, in this paper, we build on previous work to derive non-asymptotic functional approximation error bounds between the algorithm sample paths and the Ornstein-Uhlenbeck approximation using an infinite-dimensional version of Stein's method of exchangeable pairs. We show that this bound implies weak convergence under modest additional assumptions and leads to a bound on the error of the variance of the iterate averages of the algorithm. Furthermore, we use our main result to construct error bounds in terms of two common metrics: the Lévy-Prokhorov and bounded Wasserstein distances. Our results provide a foundation for developing similar error bounds for the multivariate setting and for more sophisticated stochastic approximation algorithms.

stat.ML↗

arXiv2024

Quantitative fluctuation analysis of multiscale diffusion systems via Malliavin calculus

We study fluctuations of small noise multiscale diffusions around their homogenized deterministic limit. We derive quantitative rates of convergence of the fluctuation processes to their Gaussian limits in the appropriate Wasserstein metric requiring detailed estimates of the first and second order Malliavin derivative of the slow component. We study a fully coupled system and the derivation of the quantitative rates of convergence depends on a very careful decomposition of the first and second Malliavin derivatives of the slow and fast component to terms that have different rates of convergence depending on the strength of the noise and timescale separation parameter.

math.PR↗

arXiv2023

Spherical Poisson Waves

We introduce a model of Poisson random waves in $\mathbb{S}^{2}$ and we study Quantitative Central Limit Theorems when both the rate of the Poisson process and the energy (i.e., frequency) of the waves (eigenfunctions) diverge to infinity. We consider finite-dimensional distributions, harmonic coefficients and convergence in law in functional spaces, and we investigate carefully the interplay between the rates of divergence of eigenvalues and Poisson governing measures.

math.PR↗

arXiv2023

Functional Gaussian approximations on Hilbert-Poisson spaces

We develop a functional Stein-Malliavin method in a non-diffusive Poissonian setting, thus obtaining a) quantitative central limit theorems for approximation of arbitrary non-degenerate Gaussian random elements taking values in a separable Hilbert space and b) fourth moment bounds for approximating sequences with finite chaos expansion. Our results rely on an infinite-dimensional version of Stein's method of exchangeable pairs combined with the so-called Gamma calculus. Two applications are included: Brownian approximation of Poisson processes in Besov-Liouville spaces and a functional limit theorem for an edge-counting statistic of a random geometric graph.

math.PR↗

arXiv2023

Moderate deviation principle for multiscale systems driven by fractional Brownian motion

In this paper we study the moderate deviations principle (MDP) for slow-fast stochastic dynamical systems where the slow motion is governed by small fractional Brownian motion (fBm) with Hurst parameter $H\in(1/2,1)$. We derive conditions on the moderate deviations scaling and on the Hurst parameter $H$ under which the MDP holds. In addition, we show that in typical situations the resulting action functional is discontinuous in $H$ at $H=1/2$, suggesting that the tail behavior of stochastic dynamical systems perturbed by fBm can have different characteristics than the tail behavior of such systems that are perturbed by standard Brownian motion.

math.PR↗

arXiv2021

Discrete-time inference for slow-fast systems driven by fractional Brownian motion

We study statistical inference for small-noise-perturbed multiscale dynamical systems where the slow motion is driven by fractional Brownian motion. We develop statistical estimators for both the Hurst index as well as a vector of unknown parameters in the model based on a single time series of observations from the slow process only. We prove that these estimators are both consistent and asymptotically normal as the amplitude of the perturbation and the time-scale separation parameter go to zero. Numerical simulations illustrate the theoretical results.

math.ST↗

arXiv2020

Approximation of Hilbert-valued Gaussians on Dirichlet structures

We introduce a framework to derive quantitative central limit theorems in the context of non-linear approximation of Gaussian random variables taking values in a separable Hilbert space. In particular, our method provides an alternative to the usual (non-quantitative) finite dimensional distribution convergence and tightness argument for proving functional convergence of stochastic processes. We also derive four moments bounds for Hilbert-valued random variables with possibly infinite chaos expansion, which include, as special cases, all finite-dimensional four moments results for Gaussian approximation in a diffusive context proved earlier by various authors. Our main ingredient is a combination of an infinite-dimensional version of Stein's method as developed by Shih and the so-called Gamma calculus. As an application, rates of convergence for the functional Breuer-Major theorem are established.

math.PR↗

arXiv2020

High dimensional regimes of non-stationary Gaussian correlated Wishart matrices

We study the high-dimensional asymptotic regimes of correlated Wishart matrices $d^{-1}\mathcal{Y}\mathcal{Y}^T$, where $\mathcal{Y}$ is a $n\times d$ Gaussian random matrix with correlated and non-stationary entries. We prove that under different normalizations, two distinct regimes emerge as both $n$ and $d$ grow to infinity. The first regime is the one of central convergence, where the law of the properly renormalized Wishart matrices becomes close in Wasserstein distance to that of a Gaussian orthogonal ensemble matrix. In the second regime, a non-central convergence happens, and the law of the normalized Wishart matrices becomes close in Wasserstein distance to that of the so-called Rosenblatt-Wishart matrix recently introduced by Nourdin and Zheng. We then proceed to show that the convergences stated above also hold in a functional setting, namely as weak convergence in $C([a,b];M_n(\mathbb{R}))$. As an application of our main result (in the central convergence regime), we show that it can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. Our findings complement and extend a rich collection of results on the study of the fluctuations of Gaussian Wishart matrices, and we provide explicit examples based on Gaussian entries given by normalized increments of a bi-fractional or a sub-fractional Brownian motion.

math.PR↗

arXiv2020

Typical dynamics and fluctuation analysis of slow-fast systems driven by fractional Brownian motion

This article studies typical dynamics and fluctuations for a slow-fast dynamical system perturbed by a small fractional Brownian noise. Based on an ergodic theorem with explicit rates of convergence, which may be of independent interest, we characterize the asymptotic dynamics of the slow component to two orders (i.e., the typical dynamics and the fluctuations). The limiting distribution of the fluctuations turns out to depend upon the manner in which the small-noise parameter is taken to zero relative to the scale-separation parameter. We study also an extension of the original model in which the relationship between the two small parameters leads to a qualitative difference in limiting behavior. The results of this paper provide an approximation, to two orders, to dynamical systems perturbed by small fractional Brownian noise and subject to multiscale effects.

math.PR↗

arXiv2020

Limiting behavior of large correlated Wishart matrices with chaotic entries

We study the fluctuations, as $d,n\to \infty$, of the Wishart matrix $\mathcal{W}_{n,d}= \frac{1}{d} \mathcal{X}_{n,d} \mathcal{X}_{n,d}^{T} $ associated to a $n\times d$ random matrix $\mathcal{X}_{n,d}$ with non-Gaussian entries. We analyze the limiting behavior in distribution of $\mathcal{W}_{n,d}$ in two situations: when the entries of $\mathcal{X}_{n,d}$ are independent elements of a Wiener chaos of arbitrary order and when the entries are partially correlated and belong to the second Wiener chaos. In the first case, we show that the (suitably normalized) Wishart matrix converges in distribution to a Gaussian matrix while in the correlated case, we obtain its convergence in law to a diagonal non-Gaussian matrix. In both cases, we derive the rate of convergence in the Wasserstein distance via Malliavin calculus and analysis on Wiener space.

math.PR↗

arXiv2019

Freeness characterizations on free chaos spaces

This paper deals with characterizing the freeness and asymptotic freeness of free multiple integrals with respect to a free Brownian motion or a free Poisson process. We obtain three characterizations of freeness, in terms of contraction operators, covariance conditions, and free Malliavin gradients. We show how these characterizations can be used in order to obtain limit theorems, transfer principles, and asymptotic properties of converging sequences.

math.OA↗

arXiv2018

Four moments theorems on Markov chaos

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions. For elements of a Markov chaos, this bound can be reduced to just the first four moments.

math.PR↗