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 $\partial _{t}u=-(-\Delta )^{\frac {1}{2}}u+\xi u$, where $\xi $ is a periodic spatial white noise. To be precise, we construct limits as $\varepsilon \to 0$ of solutions of $\partial _{t}u_{\varepsilon }=-(-\Delta )^{\frac {1}{2}}u_{\varepsilon }+(\xi _{\varepsilon }-C_{\varepsilon })u_{\varepsilon }$, where $\xi _{\varepsilon }$ is a mollification of $\xi $ at scale $\varepsilon $ and $C_{\varepsilon }$ 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 $\mathbf {R}^{2}$.”