Stochastic PDEs with multiscale structure (Q456194)

The authors study the spatial homogenisation of parabolic, linear stochastic partial differential equations (SPDE) of the form NEWLINE\[NEWLINE du_{\varepsilon}(x,t)=\mathcal{L}_{\varepsilon}u_{\varepsilon}(x,t)dt+\sum_{k\in\mathbb{Z}}q_{k}\left(\frac{x}{\varepsilon}\right)e^{ikx}dW_{k}(t)\quad\text{on}\;[0,2\pi] NEWLINE\]NEWLINE with homogeneous Dirichlet boundary conditions, where NEWLINE\[NEWLINE \mathcal{L}_{\varepsilon}=\frac{1}{\varepsilon}b\left(\frac{x}{\varepsilon}\right)\partial_{x}+\frac{1}{2}\sigma^{2}\left(\frac{x}{\varepsilon}\right)\partial_{x}^{2} NEWLINE\]NEWLINE and \(b,\sigma\) are suitable smooth periodic functions on \([0,2\pi]\). Assuming uniform ellipticity of \(\sigma\) and the centering condition for \(b,\sigma\), it is shown that \(u_{\varepsilon}\) converges to solutions of appropriate SPDE with diffusion coefficients depending on the the behavior of \(q_{k}\). It is pointed out that, in general, the resulting limiting SPDE does not coincide with the intuitive guess based on the corresponding deterministic homogenisation problem.