Degenerate parabolic stochastic partial differential equations

4318 M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336
hence J 1 = H + R 1 and J 2 = H + R 2 where
t
H = E (∇ x u ε ) ∗ σ (x)σ (y)(∇ y u ε
)ϱ τ (x − y)ψ δ u ε (x) − u ε (y) dxdydr
0 (T N )
2 t
R 1 = E (∇ x u ε ) ∗
σ (x)Θ δ u ε (x) − u ε (y)
0
(T N ) σ 2
× (y) − σ (x) (∇ϱτ )(x − y) − div σ (y)
ϱ τ (x − y) dxdydr
t
R 2 = E (∇ y u ε ) ∗
σ (y)Θ δ u ε (x) − u ε (y)
0
(T N ) σ 2
× (x) − σ (y) (∇ϱτ )(x − y) + div σ (x)
ϱ τ (x − y) dxdydr.
As a consequence, we see that J ε = J 4 + J 5 + R 1 + R 2 where
t
J 5 = − E σ (x)∇x u ε − σ (y)∇ y u ε 2
ϱ t (x − y)ψ δ u ε (x) − u ε (y) dxdydr
0 (T N ) 2
and therefore J ε ≤ R 1 + R 2 . Let us introduce an auxiliary function
T δ (ξ) =
ξ
0
Θ δ (ζ ) dζ.
With this in hand we obtain
t
R 1 = E σ (x)∇ x T δ u ε (x) − u ε (y)
0 (T N )
2 σ
× (y) − σ (x) (∇ϱτ )(x − y) − div σ (y)
ϱ τ (x − y) dxdydr
t
= − E T δ u ε (x) − u ε (y) div σ (x) σ (y) − σ (x) (∇ϱ τ )(x − y)
0 (T N ) 2
and similarly
R 2 = E
− σ (x)div σ (x) (∇ϱ τ )(x − y) + σ (x) σ (y) − σ (x) (∇ 2 ϱ τ )(x − y)
− div σ (x) div σ (y) ϱ τ (x − y) − σ (x)div σ (y)
(∇ϱ τ )(x − y) dxdydr
t
0
(T N ) 2 T δ
u ε (x) − u ε (y) div σ (y) σ (x) − σ (y) (∇ϱ τ )(x − y)
− σ (y)div σ (y) (∇ϱ τ )(x − y) − σ (y) σ (x) − σ (y) (∇ 2 ϱ τ )(x − y)
+ div σ (y) div σ (x) ϱ τ (x − y) − σ (y)div σ (x)
(∇ϱ τ )(x − y) dxdydr;
hence
t
R 1 + R 2 = E T δ u ε (x) − u ε (y)
0 (T N )
2
× 2 div σ (x) + div σ (y) σ (x) − σ (y) (∇ϱ τ )(x − y)
+ σ (x) − σ (y) 2 (∇ 2 ϱ τ )(x − y) + 2div σ (x) div σ (y)
ϱ τ (x − y) dxdydr.