Research interests.

 

1. Convergence to equilibrium and functional inequalities for interacting particle systems.

Markovian dynamics with many degrees of freedom are often use to sample from distributions of finite but large sets. It is therefore relevant to have quantitative estimates on the rate of convergence to equilibrium of these dynamics. One important tool to obtain this rate is provided by functional inequalities (Poincar, logarithmic-Sobolev, Nash,). Sharp estimates for the best constants in these inequalities allow in particular to obtain the dependence of the rate of convergence on the number of degrees of freedom. I have obtained, together with several colleagues and Ph.D. students but especially with Prof. G. Posta of Politectico di Milano, various estimates on the Poincar and logarithmic-Sobolev inequalities for Glauber and exchange dynamics [1, 5, 6, 12]. In particular we solved a conjecture concerning the diffusing scaling of the logarithmic-Sobolev constant for zero-range processes [5,6].

 

2. Complex systems in economics and finance.

This research has recently developed through a collaboration with Prof. W.J. Runggaldier of the University of Padova. The original motivation was to formulate and study models that could exhibit contagious default in a network of firms, and the related credit risk problem. We have proposed models with mean-field interaction, where the dynamical contagion is due to endogenous mechanisms rather than to exogenous macroeconomic factors. This is a long-term project, which involves also graduate students [46].

 

3. Stochastic control.

I have recently worked on ergodic stochastic control, in particular on almost sure optimality [19,21], also with applications to finance [9]. I have also proved the existence of the value function as a viscosity solution of the associated Bellman equation, for problem with discontinuous data [47].

 

4. Other applications of particle systems.

Interacting particle systems provide a rich and flexible class of models, that are useful in many context. In [4], in collaboration with researcher of the University of Verona, some methods for mean field interactions has been applied to a model for cellular proliferation. Similarly to what done in [46] for financial applications, we have derived macroscopic equations through probabilistic limit theorems.