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

Alexander Dunlap

Published 2020-02-17Version 1

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 $\mathbf{R}^{2}$."