Degenerate parabolic stochastic partial differential equations

M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 4307
t
−E
α dν
0
(T 1 N ) 2 R 2 x,s (ξ) dx dn 2,2(y, s, ζ )
t
≤ −E
α dν
0
(T 1 N ) 2 R 2 x,s (ξ) dx dn 2,1(y, s, ζ )
and by symmetry
−E
t (T N ) 2
0
≤ −E
R
2
t (T N ) 2
0
¯ f − 2 ∂ ξ α dm 1 (x, s, ξ) dζ dy
R 2 α dν 2 y,s (ζ ) dy dn 1,1(x, s, ξ).
Thus, the desired estimate (17) follows.
In the case of f − 1 , ¯ we take t n ↑ t, write (17) for f + 1 (t n), f ¯ + 2 (t n) and let n → ∞.
f − 2
Theorem 3.3 (Comparison Principle). Let u be a kinetic solution to (1). Then there exist u + and
u − , representatives of u, such that, for all t ∈ [0, T ], f ± (x, t, ξ) = 1 u ± (x,t)>ξ for a.e. (ω, x, ξ).
Moreover, if u 1 , u 2 are kinetic solutions to (1) with initial data u 1,0 and u 2,0 , respectively, then
for all t ∈ [0, T ]
E∥u ± 1 (t) − u± 2 (t)∥ L 1 (T N ) ≤ E∥u 1,0 − u 2,0 ∥ L 1 (T N ) . (19)
Proof. Denote f 1 = 1 u1 >ξ , f 2 = 1 u2 >ξ . Let (ψ δ ), (ϱ τ ) be approximations to the identity on
R
and T N , respectively. Namely, let ψ ∈ Cc ∞ (R) be a nonnegative symmetric function satisfying
R
ψ = 1, supp ψ ⊂ (−1, 1) and set
ψ δ (ξ) = 1 δ ψ ξ
.
δ
For the space variable x ∈ T N , we employ the approximation to the identity defined in
Lemma 2.5. Then we have
E
T N
= E
f ± 1 (x, t, ξ) ¯
R
(T N ) 2
f ± 2
(x, t, ξ) dξ dx
R 2 ϱ τ (x − y)ψ δ (ξ − ζ ) f ± 1 (x, t, ξ) ¯ f ± 2 (y, t, ζ ) dξ dζ dx dy + η t(τ, δ),
where lim τ,δ→0 η t (τ, δ) = 0. With regard to Proposition 3.2, we need to find suitable bounds for
terms I, J, K.
Since b has at most polynomial growth, there exist C > 0, p > 1 such that
b(ξ) − b(ζ )
≤ Γ (ξ, ζ )|ξ − ζ |, Γ (ξ, ζ ) ≤ C
1 + |ξ| p−1 + |ζ | p−1 .
Hence
|I| ≤ E
t (T N ) 2
0
R 2
f 1 ¯ f 2 Γ (ξ, ζ )|ξ − ζ |ψ δ (ξ − ζ ) dξ dζ ∇ x ϱ τ (x − y) dx dy ds.
As the next step we apply integration by parts with respect to ζ, ξ. Focusing only on the relevant
integrals we get
f 1 (ξ) f¯
2 (ζ )Γ (ξ, ζ )|ξ − ζ |ψ δ (ξ − ζ )dζ dξ
R
R
□