On the Stochastic Heat Equation with Spatially-Colored Random forcing

TL;DR

This paper investigates the existence, uniqueness, and long-term behavior of solutions to a stochastic heat equation driven by spatially-colored Gaussian noise, including cases with measure initial data and linearized versions.

Contribution

It introduces the concept of a temperate solution for measure initial data and analyzes conditions for weak intermittency and regularity of solutions.

Findings

Unique solutions exist under certain conditions for bounded initial data.

A new concept of temperate solutions is proposed for measure initial data.

The linearized equation's solutions exhibit specific regularity properties.

Abstract

We consider the stochastic heat equation of the following form \frac{\partial}{\partial t}u_t(x) = (\sL u_t)(x) +b(u_t(x)) + \sigma(u_t(x))\dot{F}_t(x)\quad \text{for}t>0, x\in \R^d, where $\sL$ is the generator of a L\'evy 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 \emph{weakly intermittent}. In addition, we study the same equation in the case that $\mathcal{L}u$ is replaced by its massive/dispersive analogue $\mathcal{L}u-\lambda u$ where $\lambda\in\R$. Furthermore, we extend our analysis to the case that the initial data $u_0$ is a measure rather than a function. As it turns out, the stochastic PDE in question does not have a mild solution in this case. We circumvent this problem by introducing a new concept of a solution that we call a \emph{temperate solution}, and proceed to investigate the existence and uniqueness of a temperate solution. We are able to also give partial insight into the long-time behavior of the temperate solution when it exists and is unique. Finally, we look at the linearized version of our stochastic PDE, that is the case when $\sigma$ is identically equal to one [any other constant works also].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.