Analysis and Geometry of Random Fields

About the Book

This book is based on the INdAM Workshop “Analysis and Geometry of Random Fields” held in Rome, Italy, on September 4-6, 2024. Over the last decades, significant effort has been devoted to the investigation of the geometric and topological properties of random fields on manifolds, with particular emphasis on random eigenfunctions of the Laplace–Beltrami operator on Riemannian manifolds. In the spherical setting, this probabilistic model was introduced by P. Bérard in 1985 to analyze the behavior of the nodal set of the "typical" eigenfunction, in the context of S.T. Yau’s 1982 conjecture. On the two-dimensional sphere, this model finds motivations in cosmology, specifically in connection with the cosmic microwave background, and also in mathematical physics, as it admits as a scale limit (when the eigenvalue tends to infinity) the well-known Berry random wave model. The latter is a random field on the Euclidean plane that, according to M. Berry’s 1977 conjecture, should predict the local behavior of (deterministic) eigenfunctions for billiards whose dynamics are classical and chaotic. There is a growing interest in extending results on fluctuations of geometric functionals from the two-dimensional sphere to more general manifolds, higher dimensions, and broader classes of spherical random fields, including those with temporal dependence. Such time-dependent random fields are of interest for applications in several disciplines, including climate sciences and Earth sciences. Finally, in the last few years, significant attention has been directed toward the connection between neural networks and random fields. This volume collects several contributions that advance the study of these topics.

Table of Contents:

Malliavin differentiability for the excursion measure of a Gaussian field.- A note on small probabilities for spherical random fields at a critical regime.- Laguerre Expansion for Nodal Volumes and Applications.- Level area of spin random fields: a chaos decomposition.- Global universality of the expected number of zeros of non-analytic random signals.- Fractional Cointegration of Geometric Functionals.- Angular power spectrum estimation for spherical random fields using higher-order quadratic variations.- Spherical Poisson Needlets with Shrinking Bandwidth.- A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fields.- Critical Points of Random Neural Networks.

About the Author :
Anna Paola Todino is an assistant professor in Probability and Mathematical statistics at University of Eastern Piedmont, Alessandria, Italy, since June 2023. She has got her PhD in Mathematics in Natural, Social and Life Sciences at the Gran Sasso Science Institute (GSSI), L'Aquila, under the supervision of Professor Domenico Marinucci, in the 2019. During her PhD program she spent six months at the King's College of London as a visiting student. Between 2019 and 2021 she was a postdoctoral researcher at the Ruhr University of Bochum (Germany). Then she moved to Polytechnic University of Turin for one year and to University of Milano-Bicocca. From December 2022 and before moving to University of Eastern Piedmont she worked at Sapienza University of Rome as an Assistant professor. Her research topics are related to Probability Theory and Functional Analysis. More specifically she is interested in the Geometry of Random Fields and Random Wave models on manifolds, with particular interest to random fields on the sphere; constructions of spherical Wavelets/Needlets and Needlet fields; functional limit Theorems.