The stable manifold theorem for non-linear stochastic systems with memory. I. Existence of the semiflow

We consider non-linear stochastic functional differential equations (sfdeʹs) on Euclidean space. We give sufficient conditions for the sfde to admit locally compact smooth cocycles on the underlying infinite-dimensional state space. Our construction is based on the theory of finite-dimensional stochastic flows and a non-linear variational technique. In Part II of this article, the above result will be used to prove a stable manifold theorem for non-linear sfdeʹs