The aim of the project is to give substantial contribution to the understanding of evolutionary phenomena with complex features, including infinite-dimensional character and randomness. Interactions between deterministic and stochastic methods in the study of differential evolution equations will be emphasized.
The members of the research group have various expertise in these areas and continue a long established collaboration. A central point of interaction is the study of Kolmogorov equations: on one side, knowledge of their solutions gives information on the associated Markov stochastic processes and in some cases allows to prove well-posedness results for stochastic differential equations (SDEs); on the other side, SDEs and Backward SDEs provide representation formulae for the solutions to Kolmogorov equations and their non-linear variants arising e.g. in optimal control theory.
The main topics in our research program are the following.
1. Kolmogorov equations in finite and infinite dimension. On finite dimensional spaces various topics will be addressed, for instance equations with unbounded or degenerate coefficients, non-autonomous equations, invariant measures and related summability improving properties. In infinite dimension particular attention will be devoted to Dirichlet problems in subsets of Banach spaces, and to nonautonomous equations.
2. Analytic tools in infinite dimension: Sobolev and BV spaces of functions on subsets of Banach spaces endowed with a probability measure, integration by parts formulae involving surface measures, geometric measure theory etc.
3. Advanced topics in Stochastic PDEs, such as well-posedness via regularization by noise, fluid dynamics or stochastic models in cancer growth and other biological applications, stochastic PDEs on networks.
4. Optimization problems for controlled stochastic processes. We will mainly focus on nonlinear stochastic optimal control theory for infinite-dimensional systems, typically controlled stochastic PDEs. We will address the Hamilton-Jacobi-Bellman equations, methods based on backward stochastic differential equations, robustness properties, as well as applications to economic and financial problems such as optimal consumption/investment and pricing/hedging.
The project emphasizes the interdisciplinary collaboration between mathematicians with different skills, both in Analysis and in Probability. On the other hand, the members share a common background of functional-analytical tools and techniques such as semigroup theory and evolution equations in Banach spaces. They have established fruitful collaborations since many years, and a substantial interaction between the research units is expected.
Special care is devoted to introducing younger people to research. The research group already includes several Young mathematician and we plan to recruit other ones by the requested research grants.