Degenerate parabolic stochastic partial differential equations

4312 M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336
Proof. Remark, that due to the construction of f ± it holds, for all p ∈ [1, ∞),
E sup |u ± (x, t)| p dx = E sup |ξ|
0≤t≤T T N 0≤t≤T
T p dν N x,t ± (ξ) dx ≤ C. (22)
R
Now, we are able to prove that the modification u + is right-continuous in the sense of L p (T N ).
According to Proposition 3.1 applied to the solution f + , we obtain
f + (t + ε), ψ −→ f + (t), ψ , ε ↓ 0, ∀ψ ∈ L 1 (T N × R).
Setting ψ(x, ξ) = ψ 1 (x)∂ ξ ψ 2 (ξ) for some functions ψ 1 ∈ L 1 (T N ) and ∂ ξ ψ 2 ∈ Cc ∞ (R), it reads
ψ 1 (x)ψ 2 u + (x, t + ε)
dx −→ ψ 1 (x)ψ 2 u + (x, t) dx. (23)
T N T N
In order to obtain that u + (t + ε) → w u + (t) in L p (T N ), p ∈ [1, ∞), we set ψ2 δ(ξ) = ξχ δ(ξ)
where (χ δ ) is a truncation on R, i.e. we define χ δ (ξ) = χ(δξ), where χ is a smooth function
with bounded support satisfying 0 ≤ χ ≤ 1 and
⎧
⎨
1, if |ξ| ≤ 1 χ(ξ) =
2 ,
⎩
0, if |ξ| ≥ 1,
and deduce
ψ
1 (x) u + (x, t + ε) dx − ψ 1 (x) u + (x, t) dx
T N T N
≤
ψ 1 (x) u + (x, t + ε) 1 |u + (x,t+ε)|>1/2δ dx
T N
+
ψ
1 (x)ψ δ
T N 2 u + (x, t + ε) − ψ 1 (x)ψ2
δ
u + (x, t) dx
+ ψ1 (x) u + (x, t) 1|u + (x,t)|>1/2δ dx −→ 0, ε ↓ 0,
T N
since the first and the third term on the right hand side tend to zero as δ → 0 uniformly in ε due
to the uniform estimate (22) and the second one vanishes as ε → 0 for any δ by (23).
The strong convergence in L 2 (T N ) then follows easily as soon as we verify the convergence
of the L 2 (T N )-norms. This can be done by a similar approximation procedure, using ψ 1 (x) = 1
and ψ2 δ(ξ) = ξ 2 χ δ (ξ). For the strong convergence in L p (T N ) for general p ∈ [1, ∞) we employ
the Hölder inequality and the uniform bound (22).
A similar approach then shows that the modification u − is left-continuous in the sense of
L p (T N ). The rest of the proof, showing that u − (t) = u + (t) for all t ∈ [0, T ] can be carried out
similarly to [9, Corollary 12]. □
4. Existence—smooth initial data
In this section we prove the existence part of Theorem 2.10 under an additional assumption
upon the initial condition: u 0 ∈ L p (Ω; C ∞ (T N )), for all p ∈ [1, ∞). We employ the
vanishing viscosity method, i.e. we approximate Eq. (1) by certain nondegenerate problems,
while using also some appropriately chosen approximations Φ ε , B ε of Φ and B, respectively.