On the stochastic heat equation with spatially-colored random forcing

Abstract

We consider the stochastic heat equation of the following form: ∂ ∂ t u t ( x ) = ( L u t ) ( x ) + b ( u t ( x ) ) + σ ( u t ( x ) ) F ˙ t ( x ) for  t > 0 ,   x ∈ R d , \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 L \mathcal {L} is the generator of a Lévy process and F ˙ \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 u_0 is a bounded and measurable function and σ \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 L u \mathcal {L}u is replaced by its massive/dispersive analogue L u − λ u \mathcal {L}u-\lambda u , where λ ∈ R \lambda \in \mathbf {R} . We also accurately describe the effect of the parameter λ \lambda on the intermittence of the solution in the case where σ ( u ) \sigma (u) is proportional to u u [the “parabolic Anderson model”]. We also look at the linearized version of our stochastic PDE, that is, the case where σ \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.