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In this second volume, a general approach is developed to provide
approximate parameterizations of the "small" scales by the "large"
ones for a broad class of stochastic partial differential equations
(SPDEs). This is accomplished via the concept of parameterizing
manifolds (PMs), which are stochastic manifolds that improve, for a
given realization of the noise, in mean square error the partial
knowledge of the full SPDE solution when compared to its projection
onto some resolved modes. 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 a pullback limit
depending on the time-history of the modes with low wave numbers.
Non-Markovian stochastic reduced systems are then derived based on
such a PM approach. The reduced systems take the form of stochastic
differential equations involving random coefficients that convey
memory effects. The theory is illustrated on a stochastic
Burgers-type equation.
This first volume is concerned with the analytic derivation of
explicit formulas for the leading-order Taylor approximations of
(local) stochastic invariant manifolds associated with a broad
class of nonlinear stochastic partial differential equations. These
approximations take the form of Lyapunov-Perron integrals, which
are further characterized in Volume II as pullback limits
associated with some partially coupled backward-forward systems.
This pullback characterization provides a useful interpretation of
the corresponding approximating manifolds and leads to a simple
framework that unifies some other approximation approaches in the
literature. A self-contained survey is also included on the
existence and attraction of one-parameter families of stochastic
invariant manifolds, from the point of view of the theory of random
dynamical systems.
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