A Singular Parabolic Anderson Model

Abstract

We give a new example of a measure-valued process without a density, which arises from a stochastic partial differential equation with a multiplicative noise term. This process has some unusual properties. We work with the heat equation with a random potential: u_t=Delta u+kuF. Here k>0 is a small number, and x lies in d-dimensional Euclidean space with d>2. F is a Gaussian noise which is uncorrelated in time, and whose spatial covariance equals |x-y|^(-2). The exponent 2 is critical in the following sense. For exponents less than 2, the equation has function-valued solutions, and for exponents higher than 2, we do not expect solutions to exist. This model is closely related to the parabolic Anderson model; we expect solutions to be small, except for a collection of high peaks. This phenomenon is called intermittency, and is reflected in the singular nature of our process. Solutions exist as singular measures, under suitable assumptions on the initial conditions and for sufficiently small k. We investigate various properties of the solutions, such as dimension of the support and long-time behavior. As opposed to the super-Brownian motion, which satisfies a similar equation, our process does not have compact support, nor does it die out in finite time. We use such tools as scaling, self-duality and moment formulae.