We
consider
a
stochastic
wave
equation
in
space
dimension
three
driven
by a
noise
white
in
time
with
an
absolutely
continuous
correlation
measure
given
by
the
product
of a
smooth
function
and
a
Riesz
kernel.
Let
$p_{t,x}(y)$
be
the
density
of
the
law
of
the
solution
$u(t,x)$
of
such
an
equation
at
points
$(t,x)\in
]0,T]\times
\mathbb{R}^3$.
We
prove
that
the
mapping
$(t,x)\mapsto
p_{t,x}(y)$
owns
the
same
regularity
as
the
sample
paths
of
the
process
$\{u(t,x),(t,x)\in
]0,T]\times
\mathbb{R}^3\}$.
The
proof
uses
Malliavin
calculus,
in
particular
Watanabe's
integration
by
parts
formula
and
estimates
derived
from
it.
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this
abstract
in
pdf
format.