Introduction The construction of local stable and unstable manifolds near hyperbolic equi- libria is a fundamental problem in deterministic and stochastic dynamical systems. The significance of these invariant manifolds consists in a characterization of the local behavior of the dynamical system in terms of longtime asymptotics of its tra- jectories near a stationary point. In recent years, it has been established that local stable/unstable manifolds exist for finite-dimensional stochastic ordinary differen- tial equations (sode's) ([M-S.2]) and stochastic systems with finite memory (viz. stochastic functional differential equations (sfde's))([M-S.l]). On the other hand, existence of such manifolds for nonlinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) has been an open problem since the early nineties ([F-S], [B-F], [B-F.l]). In [F-S], the existence of a random evolution operator and its Lyapunov spec- trum is established for a linear stochastic heat equation on a bounded Euclidean domain, driven by finite-dimensional white noise. For linear see's with finite- dimensional white noise, a stochastic semi-flow (i.e. random evolution operator) was obtained in [B-F]. Subsequent work on the dynamics of nonlinear spde's has focused mainly on the question of existence of continuous semiflows and the ex- istence and uniqueness of invariant measures and/or stationary solutions. Recent results on the existence of global invariant, stable/unstable manifolds (through a fixed point) for semilinear see's are given in ([D-L-S.l],[D-L-S.2]). The results in ([D- L-S.l], [D-L-S.2]) assume that the see is driven by multiplicative one-dimensional Brownian motion, with the nonlinear term having a global Lipschitz constant that is sufficiently small relative to the spectral gaps of the second-order linear operator. The latter spectral gap condition in ([D-L-S.l], [D-L-S.2]) is dictated by the use of the contraction mapping theorem. The main objective of this article is to establish the existence of local stable and unstable manifolds near stationary solutions of semilinear stochastic evolu- tion equations (see's) and stochastic partial differential equations (spde's). Our approach consists in the following two major undertakings: A construction of a sufficiently Frechet differentiable, locally compact cocycle for mild/weak trajectories of the see or the spde. Part 1 of this paper is devoted to detailing the construction of the cocycle. The application of classical nonlinear ergodic theory techniques developed by Oseledec [O] and Ruelle [Ru.2] in order to study the local structure of the above cocycle in a neighborhood of a hyperbolic stationary point. This struc- ture characterizes-via stable/unstable manifolds-the asymptotic stability of the cocycle near the stationary point. 1
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