Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities

Abstract

In this article we derive rigorously amplitude equations for stochastic PDEs with quadratic nonlinearities, under the assumption that the noise acts only on the stable modes and for an appropriate scaling between the distance from bifurcation and the strength of the noise. We show that, due to the presence of two distinct timescales in our system, the noise (which acts only on the fast modes) gets transmitted to the slow modes and, as a result, the amplitude equation contains both additive and multiplicative noise. As an application we study the case of the one dimensional Burgers equation forced by additive noise in the orthogonal subspace to its dominant modes. The theory developed in the present article thus allows to explain theoretically some recent numerical observations from [Rob03].

Amplitude equations for SPDEs with cubic nonlinearities

“…As indicated for the fast OU-process in Section 2.1, we obtain additional Martingale terms that lead to additive noise in an equation for the higher order correction of the amplitude, but the strength of the noise depends on the first order approximation. Unfortunately, as we rely on a Martingale representation argument of [2], we are limited in the final argument to onedimensional dominant spaces, i.e. dim N = 1.…”

“…Let b be a solution of the amplitude equation (9) and a as defined in (2). If the initial conditions satisfy a(0) = b(0), then…”

“…In order to get an equation for the higher order terms, we need to approximate martingale term in the equation for a in order to have explicit error bounds. We rely on Lemma 6.1 from [2], which is based on the martingale representation theorem. Thus we are limited in the final argument to dim N = 1.…”

“…Recently the impact of degenerate noise not acting directly on the dominant pattern was studies for equations of Burgers type formally [10] and later rigorously [2]. Here noise is transported via nonlinear interaction to the dominant modes.…”

“…Let us state without proof Lemma 6.1 from [2] to bound M a1 (T ) −M a1 (T ) where the martingale M a1 (T ) is defined in (63) and the martingaleM a1 (T ) is defined in (13).…”