On the stochastic heat equation with spatially-colored random forcing

We consider the stochastic heat equation of the following form:  where L is the generator of a L ́evy process and ̇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 u0 is a bounded and measurable function and σ 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 Lu is replaced by its massive/dispersive analogue Lu - λu, where λ ∈ R. We also accurately describe the effect of the parameter λ on the intermittence of the solution in the case where σ(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 σ 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.