Stochastic Modulation Equations on Unbounded Domains

Abstract

We study the impact of small additive space-time white noise on nonlinear SPDEs on unbounded domains close to a bifurcation, where an infinite band of eigenvalues changes stability due to the unboundedness of the underlying domain. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, and we rely on the approximation via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. One technical problem for establishing error estimates in the stochastic case rises from the spatially translation invariant nature of space-time white noise on unbounded domains, which implies that at any time the error is always very large somewhere far out in space. Thus we have to work in weighted spaces that allow for growth at infinity. As a first example we study the stochastic one-dimensional Swift-Hohenberg equation on the whole real line. In this setting, because of the weak regularity of solutions, the standard methods for deterministic modulation equations fail, and we need to develop new tools to treat the approximation. Using energy estimates we are only able to show that solutions of the Ginzburg-Landau equation are Holder continuous in spaces with a very weak weight, which provides just enough regularity to proceed with the error estimates.