A Singular Parabolic Anderson Model
Carl E Mueller, University of Rochester
Roger Tribe, University of Warwick
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.
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