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We introduce the notion of -completeness for a stochastic flow on manifold and prove a necessary and sufficient condition for a flow to be -complete. -completeness means that the flow is complete (i.e., exists on the given time interval) and that it belongs to some sort of -functional space, natural for manifolds where no Riemannian metric is specified.
For a given closed and translation invariant subspace of the bounded and uniformly continuous functions, we will give criteria for the existence of solutions to the equation , or of solutions asymptotically close to for the inhomogeneous differential equation , in general Banach spaces, where denotes a possibly nonlinear accretive generator of a semigroup. Particular examples for the space are spaces of functions with various almost periodicity properties and more general types of asymptotic behavior.