Stochastic Modulation Equations on Unbounded Domains

We study the impact of small additive space-time white noise on nonlinear stochastic partial differential equations 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.

We study the stochastic one-dimensional Swift-Hohenberg equation on the whole real line, where the dominant mode is well approximated by a stochastic Ginzburg-Landau equation. 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.

One main technical problem for establishing error estimates comes from the spatially translation invariant nature of the noise, which causes the error to be always very large somewhere far out in space. Thus we need to work in weighted spaces that allow for growth at infinity. Using energy estimates we are only able to show that solutions of the stochastic Ginzburg-Landau equation are Hölder-continuous in spaces with a very weak weight, which provides just enough regularity to proceed with the error estimates.