For the stochastic heat equation on the torus with coefficients growing like u|log u|^A near zero (A<1 for drift, A<1/4 for noise), a unique global strictly positive mild solution exists.
Parabolic Anderson model with colored noise on torus
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abstract
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 It\^{o} 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 H\"{o}lder 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.
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A stochastic heat equation with non-locally Lipschitz coefficients
For the stochastic heat equation on the torus with coefficients growing like u|log u|^A near zero (A<1 for drift, A<1/4 for noise), a unique global strictly positive mild solution exists.