Some questions related to invariant measures for stochastic Navier-Stokes equations

This proposal is for support of research in the area of Stochastic Partial Differential Equations (briefly SPDEs). Over the past two decades the exploration of SPDEs has become one of the most rapidly expanding areas in Probability Theory. In addition to applications to some fundamental problems in Mathematical Physics and Life Sciences, interest in such studies is motivated by a desire to understand and control the behaviour of complex systems that appear in many areas of natural and social sciences. Small random fluctuations such as thermal are present in all complex systems even if their fundamental theory is deterministic. It is also well known that many diverse deterministic environments and objects such as financial markets, insurance, internet traffic, turbulent phenomena, fluctuations of interfaces in phase transitions, spatial distribution of species and numerous others exhibit random behaviour. The theory of partial differential equations with small random perturbations is a powerful tool for studying the stability of such systems and properties of their stationary states. Importance of the theory of SPDEs may be demonstrated by the fact that at least five major international events have taken place over a period of just one year, starting March 2003. The broad aim of this research project will be to study invariant measures of deterministic partial differential equations and of random dynamical systems described by SPDEs and their small noise asymptotics. We will use these results to characterize physically relevant invariant measures and their properties. The project will be focused on some important equations of Mathematical Physics like Navier-Stokes, wave, Ginzburg-Landau and Cahn-Hilliard equations.