Parabolic Anderson model with colored noise on torus

We construct an intrinsic family of Gaussian noises on $d$-dimensional flat torus ${\mathbb{T}}^d$. It is the analogue of the colored noise on ${\mathbb{R}}^d$, and allows us to study stochastic PDEs on torus in the Itô sense in high dimensions. With this noise, we consider the parabolic Anderson model (PAM) with measure-valued initial conditions and establish some basic properties of the solution, including a sharp upper and lower bound for the moments and Hölder continuity in space and time. The study of the toy model of ${\mathbb{T}}^d$ in the present paper is a first step towards our effort in understanding how geometry and topology play an role in the behavior of stochastic PDEs on general (compact) manifolds.