ON THE STOCHASTIC HEAT EQUATION WITH SPATIALLY-COLORED RANDOM FORCING

We consider the stochastic heat equation of the following form: partial derivative/partial derivative t u(t)(x) = (Lu-t)(x) + b(u(t)(x)) + sigma(u(t)(x))(F) over dot(t)(x) for t > 0, x is an element of R-d, where L is the generator of a Levy process and (F) over dot 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 Lu is replaced by its massive/dispersive analogue Lu - lambda u, where lambda is an element of R. We also accurately describe the effect of the parameter. 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.