Non-Markovian Stochastic Reduced Equations

Abstract

In this chapter, an efficient stochastic reduction procedure is presented based on stochastic parameterizing manifolds introduced in Sects. 4.3 and 4.4. The goal here is to derive efficient reduced models to describe the main dynamical features of the amplitudes of the low modes. The reduction procedure developed here can be seen as an alternative to the nonlinear Galerkin method where the approximate inertial manifolds (AIMs) used therein are replaced here by the parameterizing manifolds. The resulting PM-based reduced equations are low-dimensional stochastic differential equations arising typically with random coefficients which convey noise-induced extrinsic memory effects expressed in terms of decay of correlations (see Lemma 5.1), making the stochastic reduced equations genuinely non-Markovian; see Eq. (5.19) below. These random coefficients involve the past of the noise path and exponentially decaying terms depending, in the self-adjoint case, on the gap between some linear combinations of the eigenvalues associated with the low modes and the eigenvalues associated with high modes. These gaps correspond exactly to those arising in the cross non-resonance conditions encountered in previous chapters; see (5.12) and Remark 5.3 below.