In this paper, we establish a necessary and sufficient condition
for the existence and regularity of the density of the solution to
a semilinear stochastic (fractional) heat equation with
measure-valued initial conditions. Under a mild cone condition for
the diffusion coefficient, we establish the smooth joint density at
multiple points. The tool we use is Malliavin calculus. The main
ingredient is to prove that the solutions to a related stochastic
partial differential equation have negative moments of all orders.
Because we cannot prove u(t, x) ? D? for measure-valued initial
data, we need a localized version of Malliavin calculus.
Furthermore, we prove that the (joint) density is strictly positive
in the interior of the support of the law, where we allow both
measure-valued initial data and unbounded diffusion coefficient.
The criteria introduced by Bally and Pardoux are no longer
applicable for the parabolic Anderson model. We have extended their
criteria to a localized version. Our general framework includes the
parabolic Anderson model as a special case.
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