EAN: 9783319125190

SPRINGERBRIEFS IN MATHEMATICS STOCHASTIC PARA...

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.

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