Modulation Equation and SPDEs on Unbounded Domains

Abstract

We consider the approximation via modulation equations for nonlinear SPDEs on unbounded domains with additive space time white noise. Close to a bifurcation an infinite band of eigenvalues changes stability, and we study the impact of small space-time white noise on this bifurcation.As a first example we study the stochastic Swift-Hohenberg equation on the whole real line. Here due to the weak regularity of solutions the standard methods for modulation equations fail, and we need to develop new tools to treat the approximation.As an additional result we sketch the proof for local existence and uniqueness of solutions for the stochastic Swift-Hohenberg and the complex Ginzburg Landau equations on the whole real line in weighted spaces that allow for unboundedness at infinity of solutions, which is natural for translation invariant noise like space-time white noise. Moreover we use energy estimates to show that solutions of the Ginzburg-Landau equation are Hölder continuous and have moments in those functions spaces. This gives just enough regularity to proceed with the error estimates of the approximation result.

Modulation and amplitude equations on bounded domains for nonlinear SPDEs driven by cylindrical α-stable Lévy processes

“…Moreover, it would be interesting to extend the present considerations to SPDEs on unbounded domains that are intensely studied over the past few years, cf. [4,11]. Such studies are currently in progress and will be reported in future publications.…”

“…Nevertheless, we will not focus on this case here. See [11] for the full approximation of Swift-Hohenberg perturbed by space-time white noise on the whole real line, [21] in the case of a simple one-dimensional noise, and [12] for large domain.…”

Amplitude equations for SPDEs driven by fractional additive noise with small Hurst parameter

“…In [7] the authors study large domains, where the dominating pattern is slowly modulated in space and the amplitude equation is thus an SPDE of Ginzburg-Landau type. See also [2] for an example of a fully unbounded domain.…”

“…Numerous extensions of our results are imaginable. While including quadratic or higher order nonlinearities seem to be technical but possible, unbounded or large domains as in [2] are a more challenging question. Moreover, it would be desirable to extend these results to SPDEs perturbed by rough nonlinear multiplicative noise.…”

The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations

“…For the case of SPDEs on unbounded domains the effective equation is no longer an SDE, but the reduced model is still given as an infinite dimensional SPDE. For details see [22] in the case of a simple one-dimensional noise, [7] for large domains and [2,19] for results with space-time white noise and on an unbounded domain. Here we will focus on the case of bounded domains only.…”