  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.

Probability;Analysis of PDEs