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 Gaussiandriving 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.
Highlights
There is a significant interest towards objects that contain one structure at a macroscopic scale, overlaying a totally different structure on a microscopic scale
To prove these three convergence results, we develop several tools that are useful when dealing with any SPDE whose underlying diffusion is driven by Lε
If we can adjust our estimates on the remainder to ensure that θ > α, so that εθ−α does decay, formally we have shown that uε(t), em is equal in distribution to a process that converges to t qρ −α e−μm2(t−s)dWm(s), which is the m-th Fourier mode of the solution to the limiting SPDE (4.4)
Summary
There is a significant interest towards objects that contain one structure at a macroscopic scale, overlaying a totally different structure on a microscopic scale. If f is sufficiently regular, it follows from (1.3) that uε → u as ε → 0, where u satisfies the PDE Such results have been widely generalised in both the forcing terms considered and the structural assumptions placed on the generator Lε, see for example [11, 4, 8, 16]. The result looks like convergence in mean-squared, it is merely disguised convergence in law since we must define the limiting solution on a different probability space to the original SPDE. Such results are often obtained artificially using the Skorokhod embedding theorem.
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