Degenerate parabolic stochastic partial differential equations

4310 M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336
t
−→ E g(y, y, s)div σ (y) dy ds
0 T N
so it remains to verify
t
−E g(x, y, s) σ (x) − σ (y) (∇ϱ τ )(x − y) dx dy ds
0 (T N )
2 t
−→ E g(y, y, s)div σ (y) dy ds.
0 T N
Here, we employ again the arguments of the commutation lemma of DiPerna and Lions (see
[10, Lemma II.1], cf. Lemma 2.5). Let us denote by g i the ith element of g and by σ i the ith
row of σ . Since τ|∇ϱ τ |(·) ≤ Cϱ 2τ (·) with a constant independent of τ, we obtain the following
estimate
t
E
g
0
T i (x, y, s) σ i (x) − σ i (y) (∇ϱ τ )(x − y) dx
dy ds
N T N σ i (x ′ ) − σ i (y ′ )
T
≤ C ess sup
x ′ ,y ′ ∈T N τ E g i (x, y, s) ϱ2τ (x − y) dx dy ds.
0 (T N ) 2
|x ′ −y ′ |≤τ
Note that according to [14,28], the square-root matrix of A is Lipschitz continuous and therefore
the essential supremum can be estimated by a constant independent of τ. Next
T
E g i (x, y, s) ϱ2τ (x − y) dx dy ds
0 (T N ) 2
T
≤ E
g i (x, y, s) 1
2 1
2
2 ϱ 2τ (x − y) dx dy ds ϱ 2τ (x − y) dx dy
0 (T N ) 2 (T N ) 2
T
≤ E (∇x u 1 )
T ∗ σ (x)
1
2
2 ϱ 2τ (x − y) dy dx ds
N T N
0
≤ (∇x u 1 ) ∗ σ (x)
L 2 (Ω×T N ×[0,T ]) .
So we get an estimate which is independent of τ and δ. It is sufficient to consider the case when
g i and σ i are smooth. The general case follows by the density argument from the above bound.
It holds
t
−E g i (x, y, s) σ i (x) − σ i (y) (∇ϱ τ )(x − y) dx dy ds
0 (T N ) 2
= − 1 t 1
τ N+1 E g
0
(T i (x, y, s) Dσ i y + r(x − y) (x − y)
N ) 2 0
x − y
·(∇ϱ) dr dx dy ds
τ
t 1
= −E
g
(T i (y + τ z, y, s) Dσ i (y + rτ z)z · (∇ϱ)(z) dr dz dy ds
N ) 2
−→ −E
0
t
0
0
(T N ) 2 g i (y, y, s) Dσ i (y)z · (∇ϱ)(z) dz dy ds.