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 partial derivative(t)u = -(-Delta)(1/2) u + xi u, where xi is a periodic spatial white noise. To be precise, we construct limits as epsilon -> 0 of solutions of partial derivative(t)u(epsilon) = -(-Delta)(1/2)u(epsilon) + (xi(epsilon) - C-epsilon)u(epsilon), where xi(epsilon) is a mollification of xi at scale epsilon and C-epsilon is a logarithmically diverging renormalization constant. We use a simple renormalization scheme based on that of Hairer and Labbe, "A simple construction of the continuum parabolic Anderson model on R-2."