The scientific activity in this area has several fruitful connections with analysis and statistics, and various applications in economics, finance, physics, biology, computer science, and other sciences. The detailed activity of the group can be found here.

The group runs a Seminar Series: more info can be found at this link.

#### Research Topics

##### Stochastic evolutions

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.

##### Members:

- Andrea Agazzi
- Francesco Grotto
- Mario Maurelli
- Marco Romito

##### Collaborators:

- Franco Flandoli (Scuola Normale Superiore, Pisa)
- Carina Geldhauser (Lund University)
- Jonathan C. Mattingly (Duke University)
- Umberto Pappalettera (Scuola Normale Superiore, Pisa)

##### Probabilistic methods in statistical learning

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.

##### Members:

- Andrea Agazzi
- Marco Romito
- Dario Trevisan

##### Collaborators:

- Jianfeng Lu (Duke University)
- Sayan Mukherjee (Max Planck Institute, Leipzig)
- Maurizio Parton (Università di Chieti-Pescara)

##### Variational problems in commutative and non-commutative probability

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.

##### Members:

- Dario Trevisan

##### Collaborators:

- Giacomo De Palma (Università di Bologna)
- Martin Huesmann (Munster University)
- Michael Goldman (CRNS Paris)
- Gian Maria Dall’Ara (INdAM)

##### Stochastic processes for chemical reactions networks

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.

##### Members:

- Andrea Agazzi

##### Collaborators:

- David Anderson (University of Wisconsin-Madison)
- Daniele Cappelletti (Politecnico di Torino)
- Jonathan C. Mattingly (Duke University)

#### Members

##### Staff

Name | Surname | Links | Personal Card |
---|---|---|---|

Andrea | Agazzi | [Mathscinet] | |

Mario | Maurelli | [arXiv] [Mathscinet] [Orcid] | |

Marco | Romito | [arXiv] [Google Scholar] [Mathscinet] [Orcid] | |

Dario | Trevisan | [Google Scholar] [Mathscinet] [Orcid] |

##### Postdoctoral Fellows

Currently, there are no postdoctoral fellows.

##### Past Members

Name | Surname | Links |
---|---|---|

Giacomo Filippo | Di Gesù | [Mathscinet] |

Franco | Flandoli | [Mathscinet] [Orcid] |

Carina | Geldhauser | [Google Scholar] [Mathscinet] [Orcid] |

Rita | Giuliano | [Mathscinet] [Orcid] |

Marta | Leocata | [Google Scholar] [Mathscinet] |

Maurizio | Pratelli | [Mathscinet] |

Maurizia | Rossi | [Google Scholar] [Mathscinet] [Orcid] |

### Grants

##### Current Grants

Currently, there are no grants.