Degenerate parabolic stochastic partial differential equations

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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 □
4308 M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 = f 1 (ξ) Γ (ξ, ζ ′ )|ξ − ζ ′ |ψ δ (ξ − ζ ′ )dζ ′ dξ R R ζ − f 1 (ξ) Γ (ξ, ζ ′ )|ξ − ζ ′ |ψ δ (ξ − ζ ′ )dζ ′ dξ dνy,s 2 (ζ ) R 2 −∞ ∞ = f 1 (ξ) Γ (ξ, ζ ′ )|ξ − ζ ′ |ψ δ (ξ − ζ ′ )dζ ′ dξ dνy,s 2 (ζ ) R 2 ζ = Υ(ξ, ζ )dν 1 R 2 x,s (ξ)dν2 y,s (ζ ) (20) where Υ(ξ, ζ ) = Therefore, we find |I| ≤ E ξ −∞ ∞ ζ t (T N ) 2 0 Γ (ξ ′ , ζ ′ )|ξ ′ − ζ ′ |ψ δ (ξ ′ − ζ ′ )dζ ′ dξ ′ . R 2 Υ(ξ, ζ ) dν 1 x,s (ξ)dν2 y,s (ζ ) ∇x ϱ τ (x − y) dx dy ds. The function Υ can be estimated using the substitution ξ ′′ = ξ ′ − ζ ′ ∞ Υ(ξ, ζ ) = Γ (ξ ′′ + ζ ′ , ζ ′ )|ξ ′′ |ψ δ (ξ ′′ ) dξ ′′ dζ ′ ζ |ξ ′′ |
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Degenerate parabolic stochastic partial differential equations

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