  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 prove a general existence and compactness theorem
for $C^k$-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. In Part
II of this work ([M-Z-Z]), we establish a local stable manifold theorem for
non-linear see's and spde's.
