The Stable Manifold Theorem for Semi-linear Stochastic Evolution Equations and Stochastic Partial Differential Equations I: the Stochastic Semiflow

The main objective of this work is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts I, II. In Part I (this paper), we establish a general existence and compactness theorem for Ck-cocycles of semilinear see’s and spde’s. Our results cover a large class of semilinear see’s as well as certain semilinear spde’s with non-Lipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinite-dimensional noise. We adopt a notion of stationarity employed in previous work of one of the authors with M. Scheutzow ([M-S.2], cf. [E-K-M-S]). In Part II of this work ([M-Z-Z]), we establish a local stable manifold theorem for non-linear see’s and spde’s.