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 $H\in(0,1)$. Close to a change of stability measured with a small parameter $\varepsilon$, we rely on the natural separation of timescales 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 OrnsteinUhlenbeck process. To this aim we need to establish an explicit averaging result for stochastic integrals driven by rough fractional noise for small Hurst parameters.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.09387
 Bibcode:
 2021arXiv210909387B
 Keywords:

 Mathematics  Probability;
 Mathematics  Dynamical Systems;
 60H15;
 60G22;
 37H20
 EPrint:
 Preprint, 32 pages