ON SOME ASYMPTOTIC RESULTS ON FUNCTIONALS OF WEAKLY STATIONARY RANDOM FIELDS

Functionals of random fields have always been a central topic in probability theory, since its inception as a subject of study. The latter include, among others, partial sums of random variables and geometric quantities associated to random functions on manifolds. In this thesis, we investigate the asymptotic probabilistic behaviour of integral functionals of weakly stationary random fields on expanding Euclidean domains, with a special focus on additive (or nonlinear) functionals of stationary Gaussian fields. In Chapter 1 we first introduce the main mathematical objects and tools encoun- tered in this work, concluding with an overview of the state of the art and our new contributions related to the main research questions of this thesis. The two main questions are the following: first, as the integration domain expands, does a central limit theorem hold? Second, given two expanding integration domains, what is the asymptotic covariance between their integral functionals? Chapter 2 contains the paper "Spectral central limit theorem for additive func- tionals of isotropic and stationary Gaussian fields", written in collaboration with Ivan Nourdin. In this chapter, we prove that a large class of additive functionals of station- ary, isotropic Gaussian fields satisfies a central limit theorem if an easily verifiable condition on the spectral measure holds. This result brings to light a new class of "strongly correlated" Gaussian fields whose additive functionals satisfy a central limit theorem. This fact contradicts the intuition forged in the last four decades, starting from the seminal works by Breuer, Dobrushin, Major, Rosenblatt and Taqqu. Chapter 3 contains the paper "Fluctuations of polyspectra in spherical and Eu- clidean random wave models", written in collaboration with Francesco Grotto and Anna Paola Todino. Our main result provides the variance rate of any additive func- tional of Euclidean (Berry’s random wave model) and spherical random waves, a problem that was left as a conjecture ten years ago. To do this, we exploit a relation between random waves and Pearson’s random walks. Chapter 4 contains the paper "Asymptotic covariances for functionals of weakly stationary random fields". Here we compute the asymptotic covariances of integral functionals of weakly stationary random fields on expanding domains under assump- tions that encompass the ones in the literature, deriving an explicit formula that involves the directional derivative of the cross covariogram of two domains. Chapter 5 contains the preprint "Limit theorems for p-domain functionals of stationary Gaussian fields", written in collaboration with Nikolai Leonenko, Ivan Nourdin and Francesca Pistolato. In this chapter we consider more general families of additive functionals, which we call p-domain functionals, including as a special case spatio-temporal functionals and 1-domain functionals considered in the previous chapters. In this setting, we are able (under suitable assumptions) to reduce the study of p-domain functionals to that of some 1-domain functionals, explaining some recent findings in the literature in a new light

Parsing patterns: emerging roles of tissue self-organization in health and disease

Patterned morphologies, such as segments, spirals, stripes, and spots, frequently emerge during embryogenesis through self-organized coordination between cells. Yet, complex patterns also emerge in adults, suggesting that the capacity for spontaneous self-organization is a ubiquitous property of biological tissues. We review current knowledge on the principles and mechanisms of self-organized patterning in embryonic tissues and explore how these principles and mechanisms apply to adult tissues that exhibit features of patterning. We discuss how and why spontaneous pattern generation is integral to homeostasis and healing of tissues, illustrating it with examples from regenerative biology. We examine how aberrant self-organization underlies diverse pathological states, including inflammatory skin disorders and tumors. Lastly, we posit that based on such blueprints, targeted engineering of pattern-driving molecular circuits can be leveraged for synthetic biology and the generation of organoids with intricate patterns

Introduzione alle misure con smartphone: esperienza di misura dell'accelerazione gravitazionale g con l'uso di un pendolo semplice

Viene descritta l’attività svolta, di supporto al progetto "LAB2GO”: aiuto alla diffusione della pratica laboratoriale nella scuola" che ha permesso di realizzare un’esperienza di Fisica per la misura dell'accelerazione gravitazionale "g" con l’uso di uno smartphone. Nella relazione vengono spiegate le tecniche sperimentali utilizzate ed indicati i risultati ottenuti effettuando misure di oscillazioni di pendoli, semplici realizzati per l'occasione, sfruttando diverse app sviluppate per smartphone. Viene infine indicata un'analisi critica dei risultati ottenuti con suggerimenti per chiunque volesse ripetere e migliorare l'esperienza

Asymptotic covariances for functionals of weakly stationary random fields

Let $(A_x)_{x\in\mathbb{R}^d}$ be a locally integrable, centered, weakly stationary random field, i.e. $\mathbb{E}[A_x]=0$, ${\rm Cov}(A_x,A_y)=K(x-y)$, $\forall x,y\in\mathbb{R}^d$, with measurable covariance function $K:\mathbb{R}^d\rightarrow\mathbb{R}$. Assuming only that $w_t:=\int_{\{|z|\le t\}}K(z)dz$ is regularly varying (which encompasses the classical assumptions found in the literature), we compute $$\lim_{t\rightarrow\infty}{\rm Cov}\left(\frac{\int_{tD}A_x dx}{t^{d/2}w_t^{1/2}}, \frac{\int_{tL}A_y dy}{t^{d/2}w_t^{1/2}}\right)$$ for $D,L\subseteq \mathbb{R}^d$ belonging to a certain class of compact sets. As an application, we combine this result with existing limit theorems to obtain multi-dimensional limit theorems for non-linear functionals of stationary Gaussian fields, in particular proving new results for the Berry's random wave model. The novel ideas of this work are mainly based on regularity conditions for (cross) covariograms of Euclidean sets and standard properties of regularly varying functions.Comment: 22 page