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

We construct solutions of a renormalized continuum fractional parabolic Anderson model, formally given by $∂_tu = −(−∆)^{1/2} u + ξu$, where $ξ$ is a periodic spatial white noise. To be precise, we construct limits as $ε → 0$ of solutions of $∂_tu_ε = −(−∆)^{1/2}u_ε + (ξ_ε − C_ε)u_ε$, where $ξ_ε$ is a mollification of $ξ$ at scale $ε$ and $C_ε$ 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 $\mathbf{R}^2$.”

Duke Scholars

Author Alexander J Dunlap Mathematics