The continuum parabolic Anderson model with a half-Laplacian and periodic noise

Abstract

We construct solutions of a renormalized continuum fractional parabolic Anderson model, formally given by ${\partial }_{t}u=-(-\Delta {)}^{1/2}u+\xi u$, where $\xi$ is a periodic spatial white noise. To be precise, we construct limits as $\epsilon \to 0$ to solutions of ${\partial }_t{u}_{\epsilon }=-(-\Delta {)}^{1/2}{u}_{\epsilon }+({\xi }_{\epsilon }-C_{\epsilon })u_{\epsilon }$, where ${\xi }_{\epsilon }$ is a mollification of $\xi$ at scale $\epsilon$ and $C_{\epsilon }$ is a logarithmically diverging renormalization constant. We use a simple renormalization scheme based on that of Hairer and Labbé, "A simple construction of the continuum parabolic Anderson model on ${\mathbf{R}}^2$."

13 pages; minor corrections in this version. To appear in Electronic Communications in Probability