Abstract
This chapter provides a general introduction to both volumes of the book. The current volume is concerned with the derivation of rigorous leading-order approximation formulas of stochastic invariant manifolds for stochastic partial differential equations (SPDEs). Volume II is concerned with extension of these formulas that provide parameterizations of the small spatial scales in terms of the large ones for SPDEs. In particular, these parameterizations lead to the effective derivation of non-Markovian reduced stochastic differential equations from Markovian SPDEs. The non-Markovian effects are here exogenous in the sense that they result from the interactions between the external driving noise and the nonlinear terms, given a projection of the dynamics onto the modes with low wave numbers. A more detailed summary of the main contributions of the current volume is given in Chap. 2.
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Notes
- 1.
Defined as graphs of random continuous functions \(h(\xi ,\omega )\) defined for each realization \(\omega \) over the whole subspace \(\fancyscript{H}^{\mathfrak {c}}\) spanned by the resolved modes, i.e., for all \(\xi \in \fancyscript{H}^{\mathfrak {c}}\), where \(\fancyscript{H}^{\mathfrak {c}}\) is typically the subspace spanned by the first few eigenmodes with low wavenumbers.
- 2.
Here \(P_{\mathfrak {c}}u\) denotes the projection of \(u\) onto the resolved modes.
- 3.
Defined as graphs over a neighborhood \(\fancyscript{V}\) contained in the subspace spanned by the critical modes, for \(\lambda \) sufficiently close to \(\lambda _c\).
- 4.
Furthermore, we mention that our particular choice of multiplicative noise allows us to consider—via the cohomology approach (see Sect. 3.3)—transformed versions (\(\omega \) by \(\omega \)) of our backward-forward systems (such as [37, system (4.3)]) so that we do not have to deal with adaptiveness issues which arise in solving more general stochastic equations backward in time [103].
- 5.
As projected onto the high modes.
- 6.
Such as the autocorrelation and probability density functions.
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Chekroun, M.D., Liu, H., Wang, S. (2015). General Introduction. In: Approximation of Stochastic Invariant Manifolds. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-12496-4_1
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DOI: https://doi.org/10.1007/978-3-319-12496-4_1
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