Let
U
a
be
the
solution
of
the
rtö
stochastic
parabolic
Cauchy
problem
Du/&t
-
Lau
=
~
.
\7u,
ult=o
=
f.
We
prove
that
U
a
depends
continuously
on
a,
when
the
coefficients
in
La
converge
to
those
in
L
o
.
This
result
is
used
to
study
the
diffusion
limit
for
the
Cauchy
problem
(in
Stratonovich
sense):
when
the
coefficients
of
La
tends
to
0
the
corresponding
solutions
U
a
converge
to
a
function
Uo
satisfying
Duo/at
=
~
0
\7uo,
ult=o
=
f.
A
criterion
is
provided
for
the
existence
of
strong
limits,
e.g.
U
a
---+
Uo,
in
the
space
of
Hida
distributions
(S)*.
As
an
application
we
show
that
weak
solutions
of
the
above
Cauchy
problem
are
strong
solutions.
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