A Singular Parabolic Anderson Model

Abstract

We consider the heat equation with a singular random potential term. The potential is Gaussian with mean 0 and covariance given by a small constant times the inverse square of the distance. Solutions exist as singular measures, under suitable assumptions on the initial conditions and for sufficiently small noise. We investigate various properties of the solutions using such tools as scaling, self-duality and moment formulae. This model lies on the boundary between nonexistence and smooth solutions. It gives a new model, other than the superprocess, which has measure-valued solutions.