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This article proposes for stochastic partial differential equations (SPDEs)driven by additive noise, a novel approach for the approximateparameterizations of the ``small'' scales by the ``large'' ones, along with thederivaton of the corresponding reduced systems. This is accomplished by seekingfor stochastic parameterizing manifolds (PMs) introduced in a previous work bythe authors(*), which are random manifolds aiming to provide --- in a meansquare sense --- such approximate parameterizations. Backward-forward systemsare designed to give access to such PMs as pullback limits depending throughthe nonlinear terms on the time-history of the dynamics of the low modes whenthe latter is simply approximated by its stochastic linear component. It isshown that the corresponding pullback limits can be efficiently determined,leading in turn to an operational procedure for the derivation of non-Markovianreduced systems able to achieve good modeling performances in practice. This isillustrated on a stochastic Burgers-type equation, where it is shown that thecorresponding non-Markovian features of these reduced systems play a key roleto reach such performances.

Non-Markovian Reduced Systems for Stochastic Partial Differential Equations: The Additive Noise Case