Stochastic integration in Banach spaces and applications to parabolic evolution equations

Stochastic partial differential equations (SPDEs) of evolution type are usually modelled as ordinary stochastic differential equations (SDEs) in an infinite-dimensional state space. In many examples such as the stochastic heat and wave equation, this viewpoint may lead to existence and uniqueness results and regularity properties. To model the equation in such a way one needs a stochastic integration theory for processes with values in an infinite-dimensional space. This approach has been considered by many authors using semigroup methods and Hilbert space methods from the 70th's up to now. There are situations where it is more natural to model the SPDE in a function space which is not a Hilbert space but only a Banach space. The main problem for this is to find a "good" stochastic integration theory for processes with values in a Banach space. In the 70th's and 80th's several authors found negative results in this direction, and it turned out that the stochastic integration theory for Hilbert spaces does not extend to the Banach space setting. We have shown that if one reformulates the integration theory for Hilbert spaces, then it does generalize to a certain class of Banach spaces. For this class of spaces we have given a complete description of the stochastically integrable processes and we have developed a stochastic calculus. The results have been applied to several SPDEs.Electrical Engineering, Mathematics and Computer Scienc