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

TL;DR

This paper derives simplified amplitude equations for SPDEs driven by very rough fractional noise with small Hurst parameter, providing a first-order approximation by an SDE and a second-order by an Ornstein-Uhlenbeck process.

Contribution

It introduces an explicit averaging method for stochastic integrals driven by rough fractional noise and establishes amplitude equations for SPDEs with small Hurst parameters.

Findings

Approximate SPDE solutions by SDEs with bounded error depending on Hurst parameter.

Developed explicit averaging results for stochastic integrals with rough fractional noise.

Provided a framework for analyzing SPDEs near stability changes with rough noise.

Abstract

We study stochastic partial differential equations (SPDEs) with potentially very rough fractional noise with Hurst parameter $H\in(0,1)$. Close to a change of stability measured with a small parameter $\varepsilon$, we rely on the natural separation of time-scales and establish a simplified description of the essential dynamics. We prove that up to an error term bounded by a power of $\varepsilon$ depending on the Hurst parameter we can approximate the solution of the SPDE in first order by an SDE, the so called amplitude equation, and in second order 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.