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 $\varepsilon\to 0$ to solutions of $\partial_t u_\varepsilon=-(-\Delta)^{1/2}u_\varepsilon+(\xi_\varepsilon-C_\varepsilon)u_\varepsilon$, 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 Labb\'e, "A simple construction of the continuum parabolic Anderson model on $\mathbf{R}^{2}$."