## 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.

### History

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- Mathematical Sciences