On the stochastic heat equation with spatially-colored random forcing

Abstract:

We consider the stochastic heat equation of the following form: \begin{equation*} \frac {\partial }{\partial t}u_t(x) = (\mathcal {L} u_t)(x) +b(u_t(x)) + \sigma (u_t(x))\dot {F}_t(x)\quad \text {for }t>0,\ x\in \mathbf {R}^d, \end{equation*} where $\mathcal {L}$ is the generator of a Lévy process and $\dot {F}$ is a spatially-colored, temporally white, Gaussian noise. We will be concerned mainly with the long-term behavior of the mild solution to this stochastic PDE.

For the most part, we work under the assumptions that the initial data $u_0$ is a bounded and measurable function and $\sigma$ is nonconstant and Lipschitz continuous. In this case, we find conditions under which the preceding stochastic PDE admits a unique solution which is also weakly intermittent. In addition, we study the same equation in the case where $\mathcal {L}u$ is replaced by its massive/dispersive analogue $\mathcal {L}u-\lambda u$, where $\lambda \in \mathbf {R}$. We also accurately describe the effect of the parameter $\lambda$ on the intermittence of the solution in the case where $\sigma (u)$ is proportional to $u$ [the “parabolic Anderson model”].

We also look at the linearized version of our stochastic PDE, that is, the case where $\sigma$ is identically equal to one [any other constant also works]. In this case, we study not only the existence and uniqueness of a solution, but also the regularity of the solution when it exists and is unique.