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 $\partial_t u=-(-Δ)^{1/2}u+ξu$, where $ξ$ is a periodic spatial white noise. To be precise, we construct limits as $\varepsilon\to 0$ to solutions of $\partial_t u_\varepsilon=-(-Δ)^{1/2}u_\varepsilon+(ξ_\varepsilon-C_\varepsilon)u_\varepsilon$, where $ξ_\varepsilon$ is a mollification of $ξ$ 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 $\mathbf{R}^{2}$."