Kolmogorov equation associated to the stochastic reflection problem ...

Proof. Let (ϕ n ) ⊂ C 1 (K) be such that
ϕ n → 0 in L 2 (K, ν), Dϕ n → F in L 2 (K, ν; H),
as n → ∞. We have to show that F = 0. Let ψ ∈ C0(K) 1 and z ∈ Q 1/2 (H).
Then by (2.10) with ϕ n ψ replacing ϕ (see Theorem 2.2) we have that
∫
∫
〈Dϕ n (x), Q 1/2 z〉 ψ(x) ν(dx) = − 〈Dψ(x), Q 1/2 z〉 ϕ n (x) ν(dx)
K
+ 1
2µ(K)
∫
Σ
∫
ϕ n (y) ψ(y) 〈n(y), Q 1/2 z〉µ Σ (dy) +
∫
= − 〈Dψ(x), Q 1/2 z〉 ϕ n (x) ν(dx) +
K
since ψ vanishes on Σ. Letting n → ∞ we find that
∫
〈F (x), Q 1/2 z〉ψ(x)µ(dx) = 0.
H
K
∫
K
K
W z (x)ϕ n (x)ψ(x)ν(dx)
W z (x)ϕ n (x)ψ(x)ν(dx),
(2.12)
This implies F = 0 in view of the arbitrariness of ψ and z (recall Lemma 2.3
and that Q 1/2 (H) is dense in H).
We shall still denote by D the closure of D and by W 1,2 (K, ν) its domain
of definition. W 1,2 (K, ν) is a Hilbert space with the scalar product
∫
〈ϕ, ψ〉 W 1,2 (K,ν) = [ϕψ + 〈Dϕ, Dψ〉]dν.
2.3 The trace of a function of W 1,2 (K, ν)
K
In order to define the trace of a function ϕ ∈ W 1,2 (K, ν) we need a technical
lemma.
Lemma 2.5. Assume that ϕ ∈ Cb 1 (H). Then the following estimate holds,
∫
∫
|Q 1/2 n(y)| 2 ϕ 2 (y)µ Σ (dy) + |x| 2 ϕ 2 (x)ν(dx)
Σ
≤ 2Tr Q
∫
K
K
∫
ϕ 2 (x)ν(dx) + 4 Tr [Q 2 ]
K
|Dϕ(x)| 2 ν(dx).
(2.13)
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