The scientific activity in this area has several fruitful connections with analysis, statistics and algebra, and various applications in economics, finance, biology and other sciences. The main topics are *stochastic evolutions* (in particular, stochastic equations in finite and infinite dimensions, particle systems), *mathematical finance* (in particular stochastic models), *probabilistic number theory* (in particular arithmetic densities). The detailed activity of the group can be found here.

**Stochastic evolutions**

Existence and regularity of densities for solutions of stochastic differential equations and stochastic PDEs. SDEs with coefficients of low regularity, PDEs with problems of uniqueness and blow-up, and corresponding regularization properties due to the presence of noise. Asymptotical statistical properties for equations from fluid dynamics driven by noise. Interacting particle systems from models in neuronal and cellular biology and in physics.**People**

Franco Flandoli [arXiv] [mathscinet]

Carina Geldhauser

Marta Leocata

Marco Romito [arXiv] [scholar] [mathscinet]**Research grants**

PRIN 2015: Deterministic and stochastic evolution equations**Collaborators**

Dirk Blömker (Augsburg)

Arnaud Debussche (ENS Bretagne)

**Variational problems in probability, statistics and mathematical finance**

Analysis on Gaussian spaces (Malliavin calculus) with applications to ordinary differential equations and PDEs. Inequalities for random variables and densities, entropic inequalities, indetermination principles, Cramér-Rao limits. Convergence of random variational problems, laws of large numbers for empirical measures systems of interacting particle systems. Application of classic and martingale optimal transport in probability and mathematical finance.**People**

Maurizio Pratelli

Dario Trevisan**Collaborators**

Giacomo De Palma (QMATH)

Martin Huesmann (Uni Bonn)

**Probability and analytic number theory**

Sub-Gaussian variables and processes. Large deviations for weighted averages of random variables and almost sure central limit theorems, with application to problems in analytic number theory. Density of sets of integers, their arithmetic, logaritmic, exponential and Beurling-Malliavin density. Benford law.**People**

Rita Giuliano [arXiv] [mathscinet]