Degenerate parabolic stochastic partial differential equations

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M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 4325 Proof. According to Proposition 4.10, there exists a set Σ ⊂ ˜Ω × T N × [0, T ] of full measure and a subsequence still denoted by {ũ n ; n ∈ N} such that ũ n (ω, x, t) → ũ(ω, x, t) for all (ω, x, t) ∈ Σ. We infer that 1ũn (ω,x,t)>ξ −→ 1ũ(ω,x,t)>ξ (40) whenever ˜P ⊗ L T N ⊗ L [0,T ] (ω, x, t) ∈ Σ; ũ(ω, x, t) = ξ = 0, where by L T N , L [0,T ] we denoted the Lebesgue measure on T N However, the set ũ D = ξ ∈ R; ˜P ⊗ L T N ⊗ L [0,T ] = ξ > 0 and [0, T ], respectively. is at most countable since we deal with finite measures. To obtain a contradiction, suppose that D is uncountable and denote ũ 1 D k = ξ ∈ R; ˜P ⊗ L T N ⊗ L [0,T ] = ξ > , k ∈ N. k Then D = ∪ k∈N D k is a countable union, so there exists k 0 ∈ N such that D k0 is uncountable. Hence ˜P ⊗ L T N ⊗ L [0,T ] ũ ∈ D ≥ ˜P ⊗ L T N ⊗ L [0,T ] ũ ∈ Dk0 = ũ 1 ˜P ⊗ L T N ⊗ L [0,T ] = ξ > = ∞ k ξ∈D k0 ξ∈D 0 k0 and the desired contradiction follows. We conclude that the convergence in (40) holds true for a.e. (ω, x, t, ξ) and obtain by the dominated convergence theorem f˜ n −→ w∗ ˜ f in L ∞ ( ˜Ω × T N × [0, T ] × R)-weak ∗ ; (41) hence (i) follows for a subsequence and the convergence at t = 0 follows by a similar approach. As the next step we shall show that the set { ˜m n ; n ∈ N} is bounded in L 2 w ( ˜Ω; M b (T N × [0, T ] × R)). With regard to the computations used in the proof of the energy inequality, we get from (28) T σ (x)∇u n T T 2 dx dt + ε n N 0 + C k≥1 T 0 ∇u n 2 dx dt ≤ C∥u 0 ∥ 2 0 T N L 2 (T N ) T u n g n T N k (x, un ) dx dβ k (t) + C G 2 0 T N n (x, un ) dx ds. Taking square and expectation and finally by the Itô isometry, we deduce Ẽ ˜m n (T N × [0, T ] × R) 2 = E m n (T N × [0, T ] × R) 2 T = E σ (x)∇u n 2 dx dt 0 T N T + ε n ∇u n 2 2 dx dt ≤ C. 0 T N
4326 M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 Thus, according to the Banach–Alaoglu theorem, (39) is obtained (up to subsequence). However, it still remains to show that the weak* limit ˜m is actually a kinetic measure. The first point of Definition 2.1 is straightforward as it corresponds to the weak*-measurability of ˜m. The second one giving the behavior for large ξ follows from the uniform estimate (28). Indeed, let (χ δ ) be a truncation on R, then it holds, for p ∈ [2, ∞), that Ẽ |ξ| p−2 d ˜m(x, t, ξ) ≤ lim inf Ẽ |ξ| p−2 χ δ (ξ) d ˜m(x, t, ξ) T N ×[0,T ]×R δ→0 T N ×[0,T ]×R = lim inf lim Ẽ |ξ| p−2 δ→0 n→∞ T N ×[0,T ]×R × χ δ (ξ) d ˜m n (x, t, ξ) ≤ C, where the last inequality follows from (28) and the sequel. As a consequence, ˜m vanishes for large ξ. The remaining requirement of Definition 2.1 follows from [6, Theorem 3.7] since for any ψ ∈ C 0 (T N × R) t −→ ψ(x, ξ) d ˜m(s, x, ξ) T N ×[0,t]×R is F˜ ⊗ B([0, T ])-measurable and ( F˜ t )-adapted for the filtration introduced below after this proof. Finally, by the same approach as above, we deduce that there exist kinetic measures õ 1 , õ 2 such that ñ n 1 w ∗ w ∗ −→ õ 1 , ñ n 2 −→ õ 2 in L 2 w ( ˜Ω; M b (T N × [0, T ] × R))-weak ∗ . Then from (28) we obtain T Ẽ σ (x)∇ũ n 2 dx dt ≤ C; 0 T N hence the application of the Banach–Alaoglu theorem yields that, up to subsequence, σ ∇ũ n converges weakly in L 2 ( ˜Ω × T N × [0, T ]). On the other hand, from the strong convergence given by Proposition 4.10 and the fact that σ ∈ W 1,∞ (T N ), we conclude using integration by parts, for all ψ ∈ C 1 (T N × [0, T ]), that T T σ (x)∇ũ n ψ(x, t) dx dt −→ σ (x)∇ũψ(x, t) dx dt, ˜P-a.s. 0 T N 0 T N Therefore σ ∇ũ n w −→ σ ∇ũ, in L 2 (T N × [0, T ]), ˜P-a.s. Since any norm is weakly sequentially lower semicontinuous, it follows for all ϕ ∈ C 0 (T N × [0, T ] × R) and fixed ξ ∈ R, ˜P-a.s., T T σ (x)∇ũ 2 ϕ 2 (x, t, ξ) dx dt ≤ lim inf σ (x)∇ũ n 2 ϕ 2 (x, t, ξ)dxdt 0 T N n→∞ 0 T N and by the Fatou lemma T T N 0 σ (x)∇ũ 2 ϕ 2 (x, t, ξ) dδũ=ξ dx dt R
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Degenerate parabolic stochastic partial differential equations

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