Titles & Abstracts

Jürgen Angst 

Title : Zeros of iterated derivatives of stationary Gaussian processes.

Abstract : We study the sequence of point processes formed by the zeros of the iterated derivatives of a stationary Gaussian process. We show that this sequence exhibit a ``crystallization'' phenomenon, the nodal sets converging to a random lattice in dimension one, or a random periodic structure in higher dimension. We also study the fluctuations around the limit. Joint work with G. Poly and G. Zheng.

Dmitry Belyaev 

Alessia Caponera 

Title: Reproducing Kernel Approach in Heterogenous Inverse Problems. 

Abstract: Many natural phenomena give rise to inverse problems where the underlying random function cannot be observed directly, but only through indirect, discretized measurements. In such settings, a key challenge is to estimate the conformational variability of these latent functions. Hence, we develop a functional framework for mean and covariance estimation under indirect observation models by formulating the forward operator as a mapping between reproducing kernel Hilbert spaces. We establish representer theorems for both first- and second-moment estimation, and introduce Mercer’s theorem tailored to inverse problems. Furthermore, we derive uniform convergence rates for our estimators under a realistic sampling scheme, capturing the statistical efficiency of our approach. Simulations in tomographic setting demonstrate the practical viability of our method. Based on a joint work with Ho Yun and Victor M. Panaretos.

Marco Carfagnini 

Titolo:  Small Fluctuations for time-dependent spherical random fields.

Abstract: The aim of this talk is to establish small ball probabilities and Chung’s law of the iterated logarithm for time-dependent Gaussian spherical random fields. Such small fluctuation rates depend on the high-frequency behaviour of the angular power spectrum. This is a joint work with Anna Paola Todino.

Lucia Celli 

Titolo:  Entropic bounds for conditionally Gaussian vectors  and applications to neural networks.

Abstract: Using entropic inequalities from information theory, we provide new bounds on the total variation and 2-Wasserstein distances between a conditionally Gaussian law and a Gaussian law with invertible covariance matrix. We apply our results  to quantify the speed of convergence to Gaussian of a randomly initialized fully connected neural network and its derivatives --- evaluated in a finite number of inputs --- when the initialization is Gaussian and the sizes of the inner layers diverge to infinity. Our results require mild assumptions on the activation function, and allow one to recover optimal rates of convergence in a variety of distances, thus improving and extending the findings of Basteri and Trevisan (2023), Favaro  et al. (2023), Trevisan (2024) and Apollonio  et al. (2024).  The talk is based on a joint work with Giovanni Peccati.

Lorenzo Cristofaro

Title: Fractional Operators in Gaussian Environment: extensions and alternatives to fractional Brownian motion. 

Abstract: We introduce new classes of Gaussian processes that exhibit distinct memory characteristics, namely Bernstein processes and Hadamard fractional Brownian motion. By applying fractional operators in a white noise framework, we are able to model a variety of memory behaviors. On one hand, these constructions yield Gaussian processes with explicit Wiener integral representations; on the other, the choice of fractional operators determines specific forms of the integrand functions. Furthermore, these processes are of Volterra type.

Lorenzo Dello Schiavo

Title: Conformally invariant random fields, quantum Liouville measures, and random Paneitz operators on Riemannian manifolds of even dimension. 

Abstract: On large classes of closed even-dimensional Riemannian manifolds M, we construct and study the Copolyharmonic Gaussian Field, i.e. a conformally invariant log-correlated Gaussian field of distributions on M. This random field is defined as the unique centered Gaussian field with covariance kernel given as the resolvent kernel of Graham—Jenne—Mason—Sparling (GJMS) operators of maximal order. The corresponding Gaussian Multiplicative Chaos is a generalization to the 2m-dimensional case of the celebrated Liouville Quantum Gravity measure in dimension two. We study the associated Liouville Brownian motion and random GJMS operator, the higher-dimensional analogues of the 2d Liouville Brownian Motion and of the random Laplacian. Finally, we study the Polyakov–Liouville measure on the space of distributions on M induced by the copolyharmonic Gaussian field, providing explicit conditions for its finiteness and computing the conformal anomaly. Based on J. London Math. Soc. (2024) https://doi.org/10.1112/jlms.70003, joint work with Ronan Herry, Eva Kopfer, Karl-Theodor Sturm.  

Simmaco Di Lillo 

Title: Spectral and Geometric Complexity of Deep Neural Networks. 

Abstract: Deep neural networks, in the infinite-width limit, give rise to isotropic Gaussian processes whose spectral and geometric features reveal much about the architecture and activation choices. In this talk, I present a unified framework based on three recent results that uncover a consistent threefold classification across different geometric descriptors of random networks. In the first work, we introduce a notion of spectral complexity and classify activation functions into three distinct regimes—sparse, low-disorder, and high-disorder—according to the asymptotic behavior of the angular power spectrum of the limiting field. This classification highlights structural differences in the expressivity of networks, with sparsity emerging prominently in deep ReLU architectures. In the second work, we study the geometry of level set boundaries. For non-smooth activations (e.g., Heaviside), we observe fractal behavior with Hausdorff dimension growing with depth. For smoother activations, the boundary volume follows one of three distinct trends—contracting, stable, or exponentially growing—mirroring the same three regimes identified via spectral analysis. In the third work, we analyze the distribution of critical points of the limiting fields. Under suitable regularity assumptions, we derive asymptotic formulas for their expected number (at fixed index or above a threshold), revealing again the same trichotomy: convergence, polynomial growth, or exponential proliferation, depending on the local behavior of the covariance kernel. The first two works are joint with Domenico Marinucci, Michele Salvi, and Stefano Vigogna.

Naomi Feldheim 

Title: Persistence and ball events for Gaussian stationary functions. 

Abstract: Let f be a Gaussian stationary function on R^d or Z^d. What is the probability that it remains above a level L, on a large ball of radius T (persistence probability)?  What is the probability of it remaining within [-L,L] on such a ball (ball probability)? In this talk, we shall see why existence of a spectral density at the origin implies existence of a "persistence exponent" (i.e.  $-lim_{T\to\infty} \log P(T)/T^d$, where $P(T)$ is the persistence probability), and why a non trivial absolutely continuous part of the spectral measure is equivalent to the positivity of the "ball exponent".  The proofs will rely on harmonic and convex analysis, and some new continuity properties. Based on joint work with Ohad Feldheim and Sumit Mukherjee (https://onlinelibrary.wiley.com/doi/10.1002/cpa.22255).

Louis Gass 

Abstract : In this talk, I will present a general method to derive lower bounds and CLTs for local functionals of Gaussian vector fields, based on a spectral analysis of the Wiener chaos decomposition. I will establish a partial equivalence between the cancellation of the first two Wiener chaoses and specific properties of the associated spectral measure. The method will be illustrated in detail through the examples of nodal volumes and critical points of Euclidean random waves. For these models, we will prove a CLT via a spectral analysis of the fourth chaos, valid in any (co)dimension. The proofs adopt a coordinate-free approach to Wiener chaos, leading to a concise and self-contained arguments with minimal computations.

Giacomo Greco 

Title: A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fields.

Abstract: We adopt a Gamma and Malliavin Calculi point of view in order to generalize Score-based diffusion Generative Models (SGMs) to an infinite-dimensional abstract Hilbertian setting. Particularly, we define the forward noising process using Dirichlet forms associated to the Cameron-Martin space of Gaussian measures and Wiener chaoses; whereas by relying on an abstract time-reversal formula, we show that the score function is a Malliavin derivative and it corresponds to a conditional expectation. This allows us to generalize SGMs to the infinite-dimensional setting. Moreover, we extend existing finite-dimensional entropic convergence bounds to this Hilbertian setting by highlighting the role played by the Cameron-Martin norm in the Fisher information of the data distribution. Lastly, we specify our discussion for spherical random fields, considering as source of noise a Whittle-Matérn random spherical field. Based on https://arxiv.org/abs/2505.13189.

Leonardo Maini 

Title: Almost sure central limit theorems via chaos expansions and related results. 

Abstract:  We investigate the asymptotic behavior of integral functionals of stationary Gaussian random fields as the integration domain tends to be the whole space. More precisely, using the Wiener chaos expansion and Malliavin-Stein method, we establish an almost sure central limit theorem (ASCLT) only under mild conditions on the covariance function of the underlying stationary Gaussian field. In this setting, we additionally derive a quantitative central limit theorem with rate of convergence in Wasserstein distance, and show certain regularity property for the said integral functionals (the latter under weaker conditions). In particular, we solved an open question on the Malliavin differentiability of the excursion volume of Berry's random wave model.

Michael McAuley 

Title: The Wiener chaos expansion for non-local functionals of Gaussian fields.

Abstract: The Wiener chaos expansion has been widely used to study local functionals of smooth Gaussian fields such as the length of their level sets. Non-local functionals, such as the number of nodal domains, have proven more challenging to analyse and generally require different techniques. In this talk I will describe recent progress in characterising the chaos expansion of such non-local functionals.

Oanh Nguyen 

Title: Random polynomials with dependent coefficients. 

Abstract: In this talk, we explore random trigonometric polynomials with dependent coefficients, moving beyond the typical assumption of independentorGaussian-distributed coefficients. We show that, under mild conditions on the dependencies between the coefficients, the asymptotic behavior of the expected number of real zeros is still universal. This universality result, to our knowledge, is the first of its kind for non-Gaussian dependent settings. Additionally, we present an elegant proof, highlighting the robustness of real zeros even in the presence of dependencies. Our findings bring the study of random polynomials closer to models encountered in practice, where dependencies between coefficients are common. Joint work with Jurgen Angst and Guillaume Poly.

Ivan Nourdin

Title: Limit theorems for the local time of Neumann-Rosenbaum fBm with Hurst parameter zero.

Abstract: In this talk, I will discuss the local time of the so-called Neumann–Rosenbaum fractional Brownian motion with Hurst parameter equal to zero. I will present results on the limiting behavior of additive functionals, obtained using a combination of techniques from Malliavin calculus, fractional calculus, and Fourier analysis. This presentation is based on ongoing joint work with Arturo Jaramillo (CIMAT, Guanajuato) and Giovanni Peccati (University of Luxembourg).

Giovanni Peccati

Title: On irreducible CLTs. 

Abstract: Central limit theorems (CLTs) for homogeneous sums (or, more generally, degenerate U-statistics) of independent random variables ubiquitous results in modern stochastic analysis, that have been recently refined and considerably extended through the use of Malliavin calculus and associated variational tools. The aim of this talk is to address the following question: 'Can one characterize those CLTs for homogeneous sums that cannot be reduced to the classical Lindberg-Feller principle?' I will provide several partial answers to this question, based on the use of Laplace spectra for (hyper)graphs, Cheeger inequalities, as well as a certain notion of "combinatorial dimension" introduced in a seminal book by R. Blei (2001). Based on a joint work with F. Caravenna and F. Cottini.

Francesca Pistolato 

Title: Uniform random waves and Berry’s cancellation. 

Abstract: Motivated by quantum chaos theory, we introduce the class of uniform random waves on compact manifolds: random linear combinations of Laplace eigenfunctions with a.s. unit L^2-norm. We provide two main results: the existence of a “chaotic” decomposition for functionals of uniform random fields; the exact cancellation of the second chaotic component for a broad class of geometric functionals. In particular, our results apply to virtually all local and non-local functionals of random spherical harmonics. To conclude, we provide a lower bound for the variance of the excursion area of uniform random waves on the 2-sphere, showing that the second chaos cancels at any level, and detect the appearance of a second-order Berry's cancellation. The talk is based on an ongoing joint work with L. Gass, D. Marinucci, G. Peccati and M. Stecconi.

Guillaume Poly 

Title: Characterization of all limits in distribution of polynomial chaoses. 

Abstract: In this talk, we will study general polynomial chaoses—namely, multivariate polynomials evaluated at random variables—and characterize their closure in distribution. In the specific case of Gaussian polynomial chaoses, this result resolves a longstanding open problem, originating from seminal works of Kolmogorov and Sevastyanov in the 1960s. We will also discuss various frameworks in which these objects arise naturally, along with several applications. Based on joint work with R.Herry & D.Malicet.

Frederic Schoppert 

Title: Directional Polynomial Wavelets on Spheres. 

Abstract: Spherical needlets, a well-established class of zonal wavelets introduced by Narcowich et al. (2006), have found many applications in the statistical analysis of spherical random fields. One of their key features is their great localization in space, which makes them a powerful tool for position-frequency investigations. However, a major drawback of spherical needlets is the fact that, due to their isotropic nature, they are not suited for the analysis of anisotropic features. For this reason, directional polynomial wavelets have been introduced by Wiaux et al. (2008) on the two-dimensional sphere. Indeed, these wavelets can be viewed as a generalization of spherical needlets and have been shown to exhibit similar localization properties (see McEwen et al. (2018)). In this talk, we present a generalized construction of directional polynomial wavelets, applicable to all higher-dimensional spheres. Therein, all essential features, in particular the localization properties, are preserved. As a consequence, the wavelet expansion admits good approximation properties, such as uniform convergence for continuous signals. The talk is based on Schoppert (2025).

Radomyra Shevchenko 

Title: Needlet-based nonparametric variance estimation on the sphere. 

Abstract: We consider a regression model at uniformly distributed locations on the unit sphere with an unknown mean function and a heteroskedastic variance. For this model, we propose an estimator for the variance function based on spherical needlets. We discuss the construction of the estimator as well as its asymptotic properties as both the number of observations and the number of the estimated needlet coefficients tend to infinity. This is ongoing joint work with Claudio Durastanti.

Michele Stecconi 

Title: Variance asymptotics in Yau’s conjecture. 

Abstract: In this talk, I approach Yau's conjecture from a probabilistic perspective by replacing deterministic eigenfunctions with Gaussian linear combinations, known as Riemannian random waves. In this setting, a natural question is the precise asymptotic behavior of the variance of the nodal volume. I will present a general strategy to study this variance based on a new chaos decomposition, which applies to random waves on general manifolds. In particular, I will discuss a version of Berry's cancellation phenomenon for monochromatic random waves and a new upper bound on the variance in the high-frequency limit for manifolds with negative curvature. (Based on a work in preparation with Anna Paola Todino)

Anna Paola Todino 

Title: New Chaos Decomposition of Gaussian Nodal Volumes 

Abstract. We investigate the random variable defined by the volume of the zero set of a smooth Gaussian field, on a general Riemannian manifold possibly with boundary. We prove a new explicit formula for its Wiener-Itô chaos decomposition that is notably simpler than existing alternatives and which holds in greater generality, without requiring the field to be compatible with the geometry of the manifold. A key advantage of our formulation is a significant reduction of the complexity of the computations of the variance of the nodal volume. Unlike the standard Hermite expansion, which requires evaluating the expectation of products of 2+2n Hermite polynomials, our approach reduces this task—in any dimension n—to computing the expectation of a product of just four Hermite polynomials. As a consequence, we establish a new exact formula for the variance. This is a joint work with Michele Stecconi.

Anna Vidotto 

Title: Functional second-order Gaussian Poincaré inequalities.

Abstract: In this paper, we work in the framework of Hilbert-valued Wiener structures and derive a functional version of the second-order Gaussian Poincaré inequality that leads to abstract bounds for Gaussian process approximation in d2 distance. Our abstract bounds are flexible and can be applied in various examples including functional Breuer-Major central limit theorems, shallow neutral networks, and stochastic partial differential equations.  

Yimin Xiao 

Title: On Geometric Properties and Extreme Probabilities of Multivariate Gaussian Random Fields.

Abstract: In this talk, we will present some results on the geometric properties and excursion probabilities of multivariate Gaussian random fields. Examples of these Gaussian random fields include those with Mat ́ern covariance/cross-covariance functions, operator fractional Brownian motion, and solutions to stochastic partial differential equations.