arXiv subjects

Luigi Amedeo Bianchi

Publications and source records attributed to Luigi Amedeo Bianchi.

15 recordsLinked to original sources

arXiv2026

MuRAL-CPD: Active Learning for Multiresolution Change Point Detection

Change Point Detection (CPD) is a critical task in time series analysis, aiming to identify moments when the underlying data-generating process shifts. Traditional CPD methods often rely on unsupervised techniques, which lack adaptability to task-specific definitions of change and cannot benefit from user knowledge. To address these limitations, we propose MuRAL-CPD, a novel semi-supervised method that integrates active learning into a multiresolution CPD algorithm. MuRAL-CPD leverages a wavelet-based multiresolution decomposition to detect changes across multiple temporal scales and incorporates user feedback to iteratively optimize key hyperparameters. This interaction enables the model to align its notion of change with that of the user, improving both accuracy and interpretability. Our experimental results on several real-world datasets show the effectiveness of MuRAL-CPD against state-of-the-art methods, particularly in scenarios where minimal supervision is available.

cs.LG↗

arXiv2025

Limit theorems for stochastic Volterra processes

We introduce an abstract Hilbert space-valued framework of Markovian lifts for stochastic Volterra equations with operator-valued Volterra kernels. Our main results address the existence and characterisation of possibly multiple limit distributions and stationary processes, a law of large numbers including a convergence rate, and the central limit theorem for time averages of the process within the Gaussian domain of attraction. As particular examples, we study Markovian lifts based on Laplace transforms in a weighted Hilbert space of densities and Markovian lifts based on the shift semigroup on the Filipović space. We illustrate our results for the case of fractional stochastic Volterra equations with additive or multiplicative Gaussian noise.

math.PR↗

arXiv2024

A Systematization of the Wagner Framework: Graph Theory Conjectures and Reinforcement Learning

In 2021, Adam Zsolt Wagner proposed an approach to disprove conjectures in graph theory using Reinforcement Learning (RL). Wagner's idea can be framed as follows: consider a conjecture, such as a certain quantity f(G) < 0 for every graph G; one can then play a single-player graph-building game, where at each turn the player decides whether to add an edge or not. The game ends when all edges have been considered, resulting in a certain graph G_T, and f(G_T) is the final score of the game; RL is then used to maximize this score. This brilliant idea is as simple as innovative, and it lends itself to systematic generalization. Several different single-player graph-building games can be employed, along with various RL algorithms. Moreover, RL maximizes the cumulative reward, allowing for step-by-step rewards instead of a single final score, provided the final cumulative reward represents the quantity of interest f(G_T). In this paper, we discuss these and various other choices that can be significant in Wagner's framework. As a contribution to this systematization, we present four distinct single-player graph-building games. Each game employs both a step-by-step reward system and a single final score. We also propose a principled approach to select the most suitable neural network architecture for any given conjecture, and introduce a new dataset of graphs labeled with their Laplacian spectra. Furthermore, we provide a counterexample for a conjecture regarding the sum of the matching number and the spectral radius, which is simpler than the example provided in Wagner's original paper. The games have been implemented as environments in the Gymnasium framework, and along with the dataset, are available as open-source supplementary materials.

cs.LG↗

arXiv2023

Limits of stochastic Volterra equations driven by Gaussian noise

We study stochastic Volterra equations in Hilbert spaces driven by cylindrical Gaussian noise. We derive a mild formulation for the stochastic Volterra equation, prove the equivalence of mild and strong solutions, the existence and uniqueness of mild solutions, and study space-time regularity. Furthermore, we establish the stability of mild solutions in $L^q(\R_+)$, prove the existence of limit distributions in the Wasserstein $p$-distance with $p \in [1,\infty)$, and characterise when these limit distributions are independent of the initial state of the process despite the presence of memory. While our techniques allow for a general class of Volterra kernels, they are particularly suited for completely monotone kernels and fractional Riemann-Liouville kernels in the full range $α\in (0,2)$.

math.PR↗

arXiv2022

A class of fractional Ornstein-Uhlenbeck processes mixed with a Gamma distribution

We consider a sequence of fractional Ornstein-Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely.

math.PR↗

arXiv2020

Linear Stochastic Dyadic model

We discuss a stochastic interacting particles' system connected to dyadic models of turbulence, defining suitable classes of solutions and proving their existence and uniqueness. We investigate the regularity of a particular family of solutions, called moderate, and we conclude with existence and uniqueness of invariant measures associated with such moderate solutions.

math.PR↗

arXiv2020

The impact of white noise on a supercritical bifurcation in the Swift-Hohenberg equation

We consider the impact of additive Gaussian white noise on a supercritical pitchfork bifurcation in an unbounded domain. As an example we focus on the stochastic Swift-Hohenberg equation with polynomial nonlinearity. Here we identify the order where small noise first impacts the bifurcation. Using an approximation via modulation equations, we provide a tool to analyse how the noise influences the dynamics close to a change of stability.

math.PR↗

arXiv2020

Stochastic Navier-Stokes Equations and Related Models

Regularization by noise for certain classes of fluid dynamic equations, a theme dear to Giuseppe Da Prato (see G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl., 2003), is reviewed focusing on 3D Navier-Stokes equations and dyadic models of turbulence.

math.PR↗

arXiv2019

Modulation equation and SPDEs on unbounded domains

We consider the approximation via modulation equations for nonlinear SPDEs on unbounded domains with additive space time white noise. Close to a bifurcation an infinite band of eigenvalues changes stability, and we study the impact of small space-time white noise on this bifurcation. As a first example we study the stochastic Swift-Hohenberg equation on the whole real line. Here due to the weak regularity of solutions the standard methods for modulation equations fail, and we need to develop new tools to treat the approximation. As an additional result we sketch the proof for local existence and uniqueness of solutions for the stochastic Swift-Hohenberg and the complex Ginzburg Landau equations on the whole real line in weighted spaces that allow for unboundedness at infinity of solutions, which is natural for translation invariant noise like space-time white noise. Moreover we use energy estimates to show that solutions of the Ginzburg-Landau equation are Hölder continuous and have moments in those functions spaces. This gives just enough regularity to proceed with the error estimates of the approximation result.

math.PR↗

arXiv2017

Structure function and fractal dissipation for an intermittent inviscid dyadic model

We study a generalization of the original tree-indexed dyadic model by Katz and Pavlović for the turbulent energy cascade of three-dimensional Euler equation. We allow the coefficients to vary with some restrictions, thus giving the model a realistic spatial intermittency. By introducing a forcing term on the first component, the fixed point of the dynamics is well defined and some explicit computations allow to prove the rich multifractal structure of the solution. In particular the exponent of the structure function is concave in accordance with other theoretical and experimental models. Moreover anomalous energy dissipation happens in a fractal set of dimension strictly less than 3.

math-ph↗

arXiv2016

Modulation Equation for SPDEs in unbounded domains with space-time white noise -- Linear Theory

We study the approximation of SPDEs on the whole real line near a change of stability via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. Due to the unboundedness of the underlying domain a whole band of infinitely many eigenfunctions changes stability. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, which is described by the modulation equation. As a first step towards a full theory of modulation equations for nonlinear SPDEs on unbounded domains, we focus, in the results presented here, on the linear theory for one particular example, the Swift-Hohenberg equation. These linear results are one of the key technical tools to carry over the deterministic approximation results to the stochastic case with additive forcing. One technical problem for establishing error estimates rises from the spatially translation invariant nature of space-time white noise on unbounded domains, which implies that at any time we can expect the error to be always very large somewhere in space.

math.PR↗

arXiv2016

Additive noise destroys the random attractor close to bifurcation

We provide an example for stabilization by noise. Our approach does not rely on monotonicity arguments due to the presence of higher order differential operators or mixing properties of the system as the noise might be highly degenerate. In the examples a scalar additive noise destroys a high-dimensional random attractor of a PDE on an unbounded domain. In the presence of small noise close to bifurcation all trajectories converge to a single stationary solution.

math.DS↗

arXiv2015

Pattern size in Gaussian fields from spinodal decomposition

We study the two-dimensional snake-like pattern that arises in phase separation of alloys described by spinodal decomposition in the Cahn-Hilliard model. These are somewhat universal patterns due to an overlay of eigenfunctions of the Laplacian with a similar wave-number. Similar structures appear in other models like reaction-diffusion systems describing animal coats' patterns or vegetation patterns in desertification. Our main result studies random functions given by cosine Fourier series with independent Gaussian coefficients, that dominate the dynamics in the Cahn-Hilliard model. This is not a cosine process, as the sum is taken over domains in Fourier space that not only grow and scale with a parameter of order $1/\varepsilon$, but also move to infinity. Moreover, the model under consideration is neither stationary nor isotropic. To study the pattern size of nodal domains we consider the density of zeros on any straight line through the spatial domain. Using a theorem by Edelman and Kostlan and weighted ergodic theorems that ensure the convergence of the moving sums, we show that the average distance of zeros is asymptotically of order $\varepsilon$ with a precisely given constant.

math.PR↗

arXiv2013

Uniqueness for an inviscid stochastic dyadic model on a tree

In this paper we prove that the lack of uniqueness for solutions of the tree dyadic model of turbulence is overcome with the introduction of a suitable noise. The uniqueness is a weak probabilistic uniqueness for all $l^2$-initial conditions and is proven using a technique relying on the properties of the $q$-matrix associated to a continuous time Markov chain.

math.PR↗

arXiv2012

A dyadic model on a tree

We study an infinite system of non-linear differential equations coupled in a tree-like structure. This system was previously introduced in the literature and it is the model from which the dyadic shell model of turbulence was derived. It mimics 3d Euler and Navier-Stokes equations in a rough approximation of a wavelet decomposition. We prove existence of finite energy solutions, anomalous dissipation in the inviscid unforced case, existence and uniqueness of stationary solutions (either conservative or not) in the forced case.

math.AP↗