Appl Math Optim 40:393–406 (1999)
1999 Springer-Verlag New York Inc.
Stability of Solutions of Parabolic PDEs with Random Drift
and Viscosity Limit
and H. Watanabe
Lehrstuhl f¨ur Mathematik V, University of Mannheim,
D-68131 Mannheim, Germany
Department of Applied Mathematics, Faculty of Science,
Okayama University of Science, Okayama, Japan
Abstract. Let u
be the solution of the Itˆo stochastic parabolic Cauchy problem
∂u/∂t − L
u = ξ ·∇u, u|
= f , where ξ is a space–time noise. We prove that u
depends continuously on α, when the coefﬁcientsin L
converge to those in L
result is used to study the diffusion limit for the Cauchy problem in the Stratonovich
sense: when the coefﬁcients of L
tend to 0 the corresponding solutions u
to the solution u
of the degenerate Cauchy problem ∂u
/∂t = ξ ◦∇u
= f .
These results are based on a criterion for the existence of strong limits in the space
of Hida distributions (S)
. As a by-product it is proved that weak solutions of the
above Cauchy problem are in fact strong solutions.
Key Words. Stochastic partial differential equations, White noise analysis, Tur-
bulent transport equation, Viscosity limit.
AMS Classiﬁcation. 60H15, 60G20.
Consider an incompressible ﬂuid with velocity ﬁeld w(t, x) = (w
(t, x ), w
(t, x ),
(t, x )) at time t ∈ R
and position x ∈ R
. The mass density u(t, x) of particles
suspended in this ﬂuid solves the Cauchy problem
− Lu =−w ·∇u, u|
= f, (1.1)
G. V˚age was supported by the DFG and T. Deck and J. Potthoff were partially supported by the DFG.