5. Stochastic Partial Differential Equations

The stochastic partial differential equations (SPDEs) considered here are stochastic evolution equations of the parabolic type. The theory of such SPDEs is complicated by different types of solution concepts and function spaces depending on the spatial regularity of the driving noise process. There is an extensive literature containing existence and uniqueness results as well as many other results. For details the reader is referred to the monographs Chow [19], Da Prato & Zabczyk [24, 25, 26], Grecksch & Tudor [39], Hairer [45], Prévot & Röckner [104], Rozovskii [108], and Walsh [123]. The main result of this chapter, Theorem 5.1 in Section 5.4, establishes existence, uniqueness, and (maximal) regularity of solutions of SPDEs with globally Lipschitz continuous coefficients. It is strongly related to the results of Kruse & Larsson [87] and van Neerven, Veraar & Weis [121]. For completeness its proof is given in the appendix.