Degenerate parabolic stochastic partial differential equations

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