Stochastic PDEs with multiscale structure

TL;DR

This paper investigates the homogenisation of stochastic PDEs with multiscale structures, revealing unexpected results when the forcing is space-time white noise, challenging existing assumptions for more regular noise.

Contribution

It provides new insights into the homogenisation process of stochastic PDEs with complex multiscale structures, especially under irregular forcing conditions.

Findings

Homogenised SPDEs can differ significantly from classical expectations.

Space-time white noise can lead to non-standard homogenised equations.

Multiscale structures influence the effective behavior of stochastic PDEs.

Abstract

We study the spatial homogenisation of parabolic linear stochastic PDEs exhibiting a two-scale structure both at the level of the linear operator and at the level of the Gaussian driving noise. We show that in some cases, in particular when the forcing is given by space-time white noise, it may happen that the homogenised SPDE is not what one would expect from existing results for PDEs with more regular forcing terms.