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Marco Carfagnini

Publications and source records attributed to Marco Carfagnini.

12 recordsLinked to original sources
arXiv2026

Berry-Heisenberg Random Waves

We construct a new family of random fields on the Heisenberg group $\mathbb{H}$, the sub-Riemannian analog of $\mathbb{R}^{n}$. These fields are generalized random eigenfunctions of the sub-Laplacian on $\mathbb{H}$, and can be viewed as the sub-Riemannian counterpart to the Berry random wave model in $\mathbb{R}^{n}$. The construction of such waves relies on the representation theory of $\mathbb{H}$, and differs from the Euclidean case because of the presence of infinite-dimensional unitary irreducible representations. This work represents a first step towards studying random waves and their geometry in sub-Riemannian spaces.

math.PR↗
arXiv2025

A note on small probabilities for spherical random fields at a critical regime

We consider time-dependent space isotropic and time stationary spherical Gaussian random fields. We establish Chung's law of the iterated logarithm and solve the small probabilities problem. Our results depend on the high-frequency behaviour of the angular power spectrum: the speed of decay of the small ball probability is faster as either the memory parameter or the space-parameter decreases.

math.PR↗
arXiv2025

On the Onsager-Machlup functional for the Brownian motion on the Heisenberg group

Onsager-Machlup functionals are used to describe the dynamics of a continuous stochastic process. For a stochastic process taking values in a Riemannian manifold, they have been studied extensively. We describe the Onsager-Machlup functional with respect to the sup norm for a hypoelliptic Brownian motion on a Heisenberg group. Unlike in the Riemannian case we do not rely on the tools from differential geometry such as comparison theorems or curvature bounds as these are not easily available in the sub-Riemannian setting. In addition, we study fine properties of trajectories of the hypoelliptic Brownian motion, including a new notion of horizontal continuous curves.

math.PR↗
arXiv2024

Dirichlet metric measure spaces: spectrum, irreducibility, and small deviations

We show that for ultracontractive irreducible Dirichlet metric measure spaces, the Dirichlet spectrum is discrete for a restriction to any connected open set without any assumption on regularity of the boundary. The main applications include small deviations for the corresponding Hunt process and large time asymptotics for the generalized heat content. Our examples include Riemannian and sub-Riemannian manifolds, as well as non-smooth and fractal spaces.

math.PR↗
arXiv2024

Onsager-Machlup functional for $\text{SLE}_κ$ loop measures

We relate two ways to renormalize the Brownian loop measure on the Riemann sphere. One by considering the Brownian loop measure on the sphere minus a small disk, known as the normalized Brownian loop measure; the other by taking the measure on simple loops induced by the outer boundary of the Brownian loops, known as Werner's measure. This result allows us to interpret the Loewner energy as an Onsager--Machlup functional for SLE$_κ$ loop measure for any fixed $κ\in (0, 4]$, and more generally, for any Malliavin--Kontsevich--Suhov loop measure of the same central charge.

math.PR↗
arXiv2024

Small fluctuations for time-dependent spherical random fields

We consider time-dependent space isotropic and time stationary spherical Gaussian random fields. We establish Chung's law of the iterated logarithm and solve the small probabilities problem. Our results depend on the high-frequency behaviour of the angular power spectrum: the speed of decay of the small ball probability is faster as either the memory parameter or the space-parameter decreases.

math.PR↗
arXiv2024

Spectral gap bounds on H-type groups

In this note we provide bounds on the spectral gap for the Dirichlet sub-Laplacians on $H$-type groups. We use probabilistic techniques and in particular small deviations of the corresponding hypoelliptic Brownian motion.

math.PR↗
arXiv2023

On the Support of a hypoelliptic diffusion on the Heisenberg group

We provide an elementary proof of the support of the law of a hypoelliptic Brownian motion on the Heisenberg group $\mathbb{H}$. We consider a control norm associated to left-invariant vector fields on $\mathbb{H}$, and describe the support in terms of the space of finite energy horizontal curves.

math.PR↗
arXiv2023

Dirichlet sub-Laplacians on homogeneous Carnot groups: spectral properties, asymptotics, and heat content

We consider sub-Laplacians in open bounded sets in a homogeneous Carnot group and study their spectral properties. We prove that these operators have a pure point spectrum, and prove the existence of the spectral gap. In addition, we give applications to the small ball problem for a hypoelliptic Brownian motion and the large time behavior of the heat content in a regular domain.

math.PR↗
arXiv2021

A functional Law of the Iterated Logarithm for weakly hypoelliptic diffusions at time zero

We study the almost sure behavior of solutions of stochastic differential equations (SDEs) as time goes to zero. Our main general result establishes a functional law of the iterated logarithm (LIL) that applies in the setting of SDEs with degenerate noise satisfying the weak Hormander condition but not the strong Hormander condition}. That is, SDEs in which the drift terms must be used in order to conclude hypoellipticity. As a corollary of this result, we obtain the almost sure behavior as time goes to zero of a given direction in the equation, even if noise is not present explicitly in that direction. The techniques used to prove the main results are based on large deviations applied to a non-trivial rescaling of the original system. In concrete examples, we show how to find the proper rescaling to obtain the functional LIL. Furthermore, we apply the main results to the problem of identifying regular points for hypoelliptic diffusions. Consequently, we obtain a control-theoretic criteria for a given point to be regular for the process.

math.PR↗