  A general approach to provide approximate parameterizations of the "small"
scales by the "large" ones, is developed for stochastic partial differential
equations driven by linear multiplicative noise. This is accomplished via the
concept of parameterizing manifolds (PMs) that are stochastic manifolds which
improve in mean square error the partial knowledge of the full SPDE solution
$u$ when compared to the projection of $u$ onto the resolved modes, for a given
realization of the noise.
  Backward-forward systems are designed to give access to such PMs in practice.
The key idea consists of representing the modes with high wave numbers (as
parameterized by the sought PM) as a pullback limit depending on the
time-history of the modes with low wave numbers.
  The resulting manifolds obtained by such a procedure are not subject to a
spectral gap condition such as encountered in the classical theory. Instead,
certain PMs can be determined under weaker non-resonance conditions.
  Non-Markovian stochastic reduced systems are then derived based on such a PM
approach. Such reduced systems take the form of SDEs involving random
coefficients that convey memory effects via the history of the Wiener process,
and arise from the nonlinear interactions between the low modes, embedded in
the "noise bath." These random coefficients follow typically non-Gaussian
statistics and exhibit an exponential decay of correlations whose rate depends
explicitly on gaps arising in the non-resonances conditions.
  It is shown on a stochastic Burgers-type equation, that such PM-based reduced
systems can achieve very good performance in reproducing statistical features
of the SPDE dynamics projected onto the resolved modes, such as the
autocorrelations and probability functions of the corresponding modes
amplitude.
