Degenerate parabolic stochastic partial differential equations

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M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 4327 T ≤ lim inf σ (x)∇ũ n→∞ 0 T n 2 ϕ 2 (x, t, ξ) dδũn =ξ dx dt, N R ˜P-a.s. In other words, this gives ñ 1 = |σ ∇ũ| 2 δũ=ξ ≤ õ 1 ˜P-a.s. hence ñ 2 = õ 2 + (õ 1 − ñ 1 ) is ˜P-a.s. a nonnegative measure and the proof is complete. □ Finally, let us define the following filtration generated by ũ, ˜W , ˜m Fˆ t = σ ϱ t ũ, ϱ t ˜W , ˜m(θψ), θ ∈ C([0, T ]), supp θ ⊂ [0, t], ψ ∈ C 0 (T N × R) and let ( F˜ t ) be its augmented filtration, i.e. the smallest complete right-continuous filtration that contains ( Fˆ t ). Then ũ is ( F˜ t )-predictable H −1 (T N )-valued processes since it has continuous trajectories. Furthermore, by the embeddings L p (T N ) ↩→ H −1 (T N ), p ∈ [2, ∞), and L 2 (T N ) ↩→ L p (T N ), p ∈ [1, 2), we conclude that, for all p ∈ [1, ∞), ũ ∈ L p ( ˜Ω × [0, T ], ˜P, dP ⊗ dt; L p (T N )), where ˜P denotes the predictable σ -algebra associated to ( F˜ t ) t≥0 . Remark, that function of ũ and ξ, is measurable with respect to ˜P ⊗ B(T N ) ⊗ B(R). 4.4. Passage to the limit ˜ f , a Borel In this paragraph we provide the technical details of the identification of the limit process with a kinetic solution. The technique performed here will be used also in the proof of existence of a pathwise kinetic solution. Theorem 4.13. The triple ( ˜Ω, F ˜ , ( F˜ t ), ˜P), ˜W , ũ is a martingale kinetic solution to the problem (1). Note, that as the set D from Lemma 4.12 is a complement of a set with zero Lebesgue measure, it is dense in [0, T ]. Let us define for all t ∈ D and some fixed ϕ ∈ Cc ∞(TN × R) M n (t) = f n (t), ϕ − f 0 , ϕ − f n (s), b n (ξ) · ∇ϕ ds 0 t − f n (s), div A(x)∇ϕ t ds − ε n f n (s), ϕ ds − 1 2 0 t 0 t δu n =ξ G 2 n , ∂ ξ ϕ ds + m n , ∂ ξ ϕ ([0, t)), n ∈ N, t ˜M n (t) = f˜ n (t), ϕ − f˜ n (0), ϕ − 0 t − f ˜ n (s), div A(x)∇ϕ t ds − ε n 0 t 0 ˜ f n (s), b n (ξ) · ∇ϕ ds 0 ˜ f n (s), ϕ ds − 1 2 δũn =ξ G 2 n , ∂ ξ ϕ ds + ˜m n , ∂ ξ ϕ ([0, t)), n ∈ N, 0 ˜M(t) = f ˜(t), ϕ − f ˜(0), ϕ t − f ˜ (s), b(ξ) · ∇ϕ ds 0 t − f ˜ (s), div A(x)∇ϕ ds − 1 t δũ=ξ G 2 , ∂ ξ ϕ ds + ˜m, ∂ ξ ϕ ([0, t)). 2 0 The proof of Theorem 4.13 is a consequence of the following two propositions. 0
4328 M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 Proposition 4.14. The process ˜W is an ( F˜ t )-cylindrical Wiener process, i.e. there exists a collection of mutually independent real-valued ( F˜ t )-Wiener processes { ˜β k } k≥1 such that ˜W = k≥1 ˜β k e k . Proof. Hereafter, fix K ∈ N, times 0 ≤ s 1 < · · · < s K ≤ s ≤ t, s, t ∈ D, continuous functions γ : C [0, s]; H −1 (T N ) × C [0, s]; U 0 −→ [0, 1], g : R K −→ [0, 1] and test functions ψ 1 , . . . , ψ K ∈ C 0 (T N × R) and θ 1 , . . . , θ K ∈ C([0, T ]) such that supp θ i ⊂ [0, s i ], i = 1, . . . , K . For notational simplicity, we write g( ˜m) instead of g ˜m(θ 1 ψ 1 ), . . . , ˜m(θ K ψ K ) and similarly g( ˜m n ) and g(m n ). By ϱ s we denote the operator of restriction to the interval [0, s] as introduced in (38). Obviously, ˜W is a U 0 -valued cylindrical Wiener process and is ( F˜ t )-adapted. According to the Lévy martingale characterization theorem, it remains to show that it is also a ( F˜ t )-martingale. It holds true Ẽ γ ϱ s ũ n , ϱ s ˜W n g( ˜m n ) ˜W n (t) − ˜W n (s) = E γ ϱ s u n , ϱ s W g(m n ) W (t) − W (s) = 0 since W is a martingale and the laws of (ũ n , ˜W n ) and (u n , W ) coincide. Next, the uniform estimate sup n∈N Ẽ∥ ˜W n (t)∥ 2 U 0 = sup E∥W (t)∥ 2 U 0 < ∞ n∈N and the Vitali convergence theorem yields Ẽ γ ϱ s ũ, ϱ s ˜W g( ˜m) ˜W (t) − ˜W (s) = 0 which finishes the proof. □ Proposition 4.15. The processes ˜M(t), ˜M 2 (t) − k≥1 t indexed by t ∈ D, are ( ˜ F t )-martingales. 0 δũ=ξ g k , ϕ t 2 dr, ˜M(t) ˜β k (t) − δũ=ξ g k , ϕ dr, Proof. All these processes are ( F˜ t )-adapted as they are Borel functions of ũ and ˜β k , k ∈ N, up to time t. For the rest, we use the same approach and notation as the one used in the previous lemma. Let us denote by ˜β k n, k ≥ 1 the real-valued Wiener processes corresponding to ˜W n , that is ˜W n = k≥1 ˜β k ne k. For all n ∈ N, the process M n = · 0 δu n =ξ Φ n (u n )dW, ϕ = k≥1 · 0 δu n =ξ g n k , ϕ dβ k (r) is a square integrable (F t )-martingale by (2) and by the fact that the set {u n ; n ∈ N} is bounded in L 2 (Ω; L 2 (0, T ; L 2 (T N ))). Therefore (M n ) 2 − k≥1 · 0 δu n =ξ g n k , ϕ 2 dr, M n β k − · 0 0 δu n =ξ g n k , ϕ dr
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Degenerate parabolic stochastic partial differential equations

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