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.