A Measure-Valued Process Related to the Parabolic Anderson Model

Abstract

We consider the following stochastic partial differential equation:

$$\frac{{\partial {{u}_{t}}}}{{\partial t}} = \Delta {{u}_{t}} + \kappa {{u}_{t}}\dot{F}(t, \cdot ).$$

Here, t ≥ 0, x ∈ R ^d with d ≥ 3, and κ > 0. $\dot{F}(t,x)$ is a generalized Gaussian process, with covariance

$$E\left[ {\dot{F}(t,x)\dot{F}(t,y)} \right] = \frac{{\delta (t - s)}}{{|x - y{{|}^{2}}}}.$$

Our solution u _t (dx) will exist as a nonnegative measure, provided u ₀ (dx) is a nonnegative measure satisfying certain properties. We discuss various properties of the solutions. Useful tools include the Feynman-Kac formula, duality, and integral equations.

The first author is supported by an NSF travel grant and an NSA grant.