Stochastic space-time systems

The project is aimed at investigating qualitative properties of stochastic infinite
dimensional systems (in particular, stochastic partial differential equations) and at research in infinite
dimensional stochastic control theory. More specifically, the following topics will
be emphasized: 1. Investigation of stochastic nonlinear wave equations, especially the problem
of regularity and uniqueness of martingale solutions, existence of weak solutions to
stochastic geometric wave equations in the energy space and large-time behaviour
of solutions. 2. The study of properties of stochastic convolutions driven by right-continuous martingales
or Lévy processes and by Gaussian processes that are not semimartingales (typically, the
fractional Brownian motion) in Hilbert and Banach spaces. 3. Infinite dimensional stochastic
control and the associated semilinear Hamilton-Jacobi-Bellman equations
in infinite dimensions, in particular ergodic and adaptive control problems, including the
corresponding ergodicity and uniform ergodicity problems for stochastic partial differential
equations. 4. The study of properties of the Brownian net and related measure-valued processes,
including the properties of the corresponding heat equations.