| Stochastic partial differential equations (SPDEs) arising in mathematical physics and applied sciences can often be formulated as ordinary stochastic differential equations in an infinite dimensional state space. In this way tools from functional analysis can be used to prove existence and uniqueness of solutions and to study their qualitative properties, such as regularity and long term behavior. In recent years there has been much interest in maximal regularity techniques and their applications to nonlinear partial differential equations. In particular, many differential operators are known to have maximal regularity. In this project we propose to develop a systematic theory of stochastic maximal regularity and to apply it to SPDEs. At present stochastic maximal regularity estimates are known only in very special situations (e.g. when the state space is a Hilbert spaces). In my PhD thesis the classical stochastic integration theory for Hilbert spaces was extended to a wide class of Banach spaces. Basic tools such as the Itô isometry and Itô's formula were extended to this setting. The first part of the research project (Project I) consists of applying this integration theory to find conditions implying stochastic maximal regularity, using recent H-infinity-calculus techniques as a basic tool. An important aspect of our approach is that it would immediately lead to a large class of operators having stochastic maximal regularity. In Project II we study stochastic maximal regularity for time-dependent operators. In Project III we will apply stochastic maximal regularity to nonlinear SPDEs arising in mathematical physics and other applied sciences such as the stochastic Navier-Stokes equations. It is expected that stochastic maximal regularity will lead to entirely new techniques for solving various classes of nonlinear SPDEs which are rather inaccesible with the existsing tools from the literature. |
