Degenerate parabolic stochastic partial differential equations

M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 4309
and similarly for the second term. Let us define
Θ δ (ξ) =
ξ
−∞
ψ δ (ζ ) dζ.
Then we have J = J 1 + J 2 + J 3 with
t
J 1 = −E (∇ x u 1 ) ∗
σ (x)σ (x)(∇ϱ τ )(x − y)Θ δ u1 (x, s) − u 2 (y, s) dxdyds,
0 (T N )
2 t
J 2 = −E (∇ y u 2 ) ∗
σ (y)σ (y)(∇ϱ τ )(x − y)Θ δ u1 (x, s) − u 2 (y, s) dxdyds,
0 (T N )
2 t
J 3 = −E |σ (x)∇x u 1 | 2 + |σ (y)∇ y u 2 | 2
ϱ τ (x − y)ψ δ u1 (x, s)
0 (T N ) 2
− u 2 (y, s) dx dy ds.
Let
t
H = E (∇ x u 1 ) ∗
σ (x)σ (y)(∇ y u 2 )ϱ τ (x − y)ψ δ u1 (x, s) − u 2 (y, s) dx dy ds.
0 (T N ) 2
We intend to show that J 1 = H + o(1), J 2 = H + o(1), where o(1) → 0 as τ → 0 uniformly in
δ, and consequently
t
J = −E
σ (x)∇ x u 1 − σ (y)∇ y u 2 2 ϱτ (x − y)
0 (T N ) 2
× ψ δ
u1 (x, s) − u 2 (y, s) dx dy ds + o(1) ≤ o(1). (21)
We only prove the claim for J 1 since the case of J 2 is analogous. Let us define
g(x, y, s) = (∇ x u 1 ) ∗ σ (x)Θ δ
u1 (x, s) − u 2 (y, s) .
Here, we employ again the assumption (ii) in Definition 2.2. Recall, that it gives us some
regularity of the solution in the nondegeneracy zones of the diffusion matrix A and hence
g ∈ L 2 (Ω × Tx N × TN y × [0, T ]). It holds
t
J 1 = −E g(x, y, s) σ (x) − σ (y) (∇ϱ τ )(x − y) dx dy ds
0
(T N ) 2
t
− E g(x, y, s)σ (y)(∇ϱ τ )(x − y) dx dy ds,
0 (T N )
2 t
H = E g(x, y, s)div y σ (y)ϱ τ (x − y) dx dy ds
0
(T N ) 2
t
= E g(x, y, s)div σ (y) ϱ τ (x − y) dx dy ds
0 (T N ) 2
t
− E g(x, y, s)σ (y)(∇ϱ τ )(x − y) dx dy ds,
0 (T N ) 2
where divergence is applied row-wise to a matrix-valued function. Therefore, it is enough to
show that the first terms in J 1 and H have the same limit value if τ → 0. For H, we obtain easily
t
E g(x, y, s)div σ (y) ϱ τ (x − y) dx dy ds
0 (T N ) 2