Probability and Mathematical Statistics

The research activity is concentrated on problems of existence, regularity, statistical asymptotic behaviour, and the analysis of relevant properties of solutions of ordinary and partial differential equations with random components, with emphasis on equations from fluid dynamics. The interest spans also ordinary differential equations with non-regular coefficients, partial differential equations without uniqueness or with blow-up, and corresponding regularization properties due to noise. Problems of interacting particle systems arising from models of cortices are also considered, as well as dynamical and asymptotic properties of piecewise deterministic Markov processes.

In recent years deep learning, through neural networks, has had an enormous impact on the development of artificial intelligence and on its possible applications. This remarkable success though has not been fully matched by theoretical results that could provide justifications of its efficacy or hints for improvements, or identify possible limitations, as well as construct methods to interpret its results. The research activity of the group in this direction is devoted to the analysis of the interaction between artificial neural networks and the theory of Gaussian processes, to the optimality and convergence of learning dynamics for neural networks, in different asymptotic regimes, to the analysis of stochastic gradient descent algorithms, to Bayes statistics for dynamical systems and reinforcement learning.

The main interests of this subject are analysis in Gaussian spaces, in particular Malliavin calculus, inequalities for random variables and their densities, entropic inequalities, and indeterminacy principles, in both commutative and non-commutative (quantum) settings. Other topics of interest are random optimal transportation problems, laws of large numbers for empirical measures of interacting particles, and combinatorial optimization problems in Euclidean random models.

The analysis is devoted to problems of large deviations for jump processes, and to structural stability conditions for their stability. The main themes in this analysis are averaging methods, duality for Markov processes, and variational inequalities. The focus is given to reaction-diffusion processes, propagation of chaos for systems of interacting particles, and to variational structures in discrete spaces, with emphasis on the out-of-equilibrium regime.

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• Convergence and Stability of Reaction and Interaction Network Dynamics (ConStRAINeD) (Prin 2022)

  Principal Investigator: Daniele Cappelletti

  Coordinator of the Research Unit: Andrea Agazzi

  Project period: Sep 28, 2023 – Sep 27, 2025

• Understanding the LEarning process of QUantum Neural networks (LeQun) (Prin 2022)

  Principal Investigator: Giacomo De Palma

  Coordinator of the Research Unit: Dario Trevisan

  Project period: Sep 28, 2023 – Sep 27, 2025

• Noise in fluid dynamics and related models (Prin 2022)

  Principal Investigator: Franco Flandoli

  Coordinator of the Research Unit: Marco Romito

  Project period: Sep 28, 2023 – Sep 27, 2025

• APRISE - Analysis and Probability in Science (Progetti di Ricerca di Ateneo (PRA) 2022-2023)

  Principal Investigator: Marco Romito

  Project period: Oct 10, 2022 – Dec 31, 2024

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