Degenerate parabolic stochastic partial differential equations

M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 4317
≤ E ϱ τ (x − y)ψ δ (ξ − ζ ) f
(T ε (x, t, ξ) f¯
ε (y, t, ζ ) dξ dζ dx dy dt + δ
N ) 2 R
2
≤ E ϱ τ (x − y)ψ δ (ξ − ζ ) f 0 (x, ξ) f¯
0 (y, ζ )dξdζ dxdy + δ + I
(T ε + J ε + K ε
N ) 2 R
2
≤ E ϱ τ (x − y) f 0 (x, ξ) f¯
0 (y, ξ) dξ dx dy + 2δ + I
(T ε + J ε + K ε ,
N ) 2 R
where I ε , J ε , K ε are defined correspondingly to I, J, K in Proposition 3.2 but using the
approximated coefficients B ε , A ε , Φ ε instead. From the same estimates as the ones used in the
proof of Theorem 3.3, we conclude
E ϱ τ (x − y) u ε (x, t) − u ε (y, t) + dx dy
(T N )
2
≤ E ϱ τ (x − y) u 0 (x) − u 0 (y) +
dxdy + 2δ + Ct δ −1 τ + δ −1 τ 2 + δ α + J ε .
(T N ) 2
In order to control the term J ε , recall that (keeping the notation from Theorem 3.3)
t
J ε = − E (∇ x u ε ) ∗
σ (x)σ (x)(∇ϱ τ )(x − y)Θ δ u ε (x) − u ε (y) dxdydr,
0 (T N )
2 t
− E (∇ y u ε ) ∗
σ (y)σ (y)(∇ϱ τ )(x − y)Θ δ u ε (x) − u ε (y) dxdydr,
0 (T N )
2 t
− E |σ (x)∇x u ε | 2 + |σ (y)∇ y u ε | 2
ϱ τ (x − y)ψ δ u ε (x) − u ε (y) dx dy dr
0 (T N )
2 t
− ε E
∇ x u ε − ∇ y u ε 2
ϱ τ (x − y)ψ δ u ε (x) − u ε (y) dxdydr
0 (T N ) 2
= J 1 + J 2 + J 3 + J 4 .
The first three terms on the above right hand side correspond to the diffusion term div(A(x)∇u ε ).
Since all u ε are smooth and hence the chain rule formula is not an issue here, J 4 is obtained after
integration by parts from similar terms corresponding to εu ε . Next, we have
J 1 = − E
t
0
t
(T N ) 2 (∇ x u ε ) ∗ σ (x)div y
σ (y)Θ δ
u ε (x) − u ε (y) ϱ τ (x − y)dxdydr
+ E (∇ x u ε ) ∗ σ (x) σ (y) − σ (x) (∇ϱ τ )(x − y)
0 (T N )
2
× Θ δ u ε (x) − u ε (y) dxdydr
and
t
J 2 = E
0
t
(T N ) 2 (∇ y u ε ) ∗ σ (y)div x
σ (x)Θ δ
u ε (x) − u ε (y) ϱ τ (x − y)dxdydr
+ E (∇ y u ε ) ∗ σ (y) σ (x) − σ (y) (∇ϱ τ )(x − y)
0 (T N )
2
× Θ δ u ε (x) − u ε (y) dxdydr;