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

Abstract

We study stochastic partial differential equations (SPDEs) with potentially very rough fractional noise with Hurst parameter [Formula: see text]. Close to a change of stability measured with a small parameter [Formula: see text], we rely on the natural separation of time-scales and establish a simplified description of the essential dynamics. Up to an error term bounded by a power of [Formula: see text] depending on the Hurst parameter we can approximate the solution of the SPDE in first order by an SDE, the so-called amplitude equation, which describes the amplitude of the dominating pattern changing stability. In second order the approximation is given by a fast infinite-dimensional Ornstein–Uhlenbeck process. To this aim, we need to establish an explicit averaging result for stochastic integrals driven by rough fractional noise for small Hurst parameters.