Maurizia Rossi

Representation of Gaussian isotropic spin random fields

We develop a technique for the construction of random fields on algebraic structures. We deal with two general situations: random fields on homogeneous spaces of a compact group and in the spin line bundles of the 2-sphere. In particular, every complex Gaussian isotropic spin random field can be represented in this way. Our construction extends P. Levy’s original idea for the spherical Brownian motion.

On Lévy's Brownian motion indexed by elements of compact groups

We investigate positive definiteness of the Brownian kernel K(x,y)=1/2(d(x,x0)+d(y,x0)-d(x,y)) on a compact group G and in particular for G=SO(n).

The Defect of Random Hyperspherical Harmonics

Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit d-sphere (

Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on S^d

d\ge 2

The Geometry of Spherical Random Fields

d≥2). We investigate the distribution of their defect, i.e., the difference between the measure of positive and negative regions. Marinucci and Wigman studied the two-dimensional case giving the asymptotic variance (Marinucci and Wigman in J Phys A Math Theor 44:355206, 2011) and a central limit theorem (Marinucci and Wigman in Commun Math Phys 327(3):849–872, 2014), both in the high-energy limit. Our main results concern asymptotics for the defect variance and quantitative CLTs in Wasserstein distance, in any dimension. The proofs are based on Wiener–Itô chaos expansions for the defect, a careful use of asymptotic results for all order moments of Gegenbauer polynomials and Stein–Malliavin approximation techniques by Nourdin and Peccati (in Prob Theory Relat Fields 145(1–2):75–118, 2009; Normal approximations with Malliavin calculus. Cambridge Tracts in Mathematics, vol 192, Cambridge University Press, Cambridge, 2012). Our argument requires some novel technical results of independent interest that involve integrals of the product of three hyperspherical harmonics.

Asymptotic distribution of nodal intersections for arithmetic random waves

We provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a d-dimensional general diffusion process X, as the conditioning time tends to 0. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift b of X. It turns out that the sharp asymptotics for the exit probability are independent of the drift b, provided it satisfies a simple condition that is always satisfied in dimension 1. On the other hand we produce an example where this assumption is not satisfied and the drift is actually influential.