Kolmogorov equation associated to the stochastic reflection problem ...

We show now that if ϕ ∈ W 2,2 (K, ν), then one can define the trace on
Σ of Dϕ. Similarly to the definition of the trace of ϕ on Σ we define
|Q 1/2 n(y)|γ(Dϕ) = lim n→∞ |Q 1/2 n(y)|γ(Dϕ N ) in L 2 (Σ, µ Σ ) for all {ϕ n } ⊂
C 2 (K), ϕ n → ϕ in W 2,2 (K, ν).
Proposition 2.9 below shows that this trace is well defined.
Proposition 2.9. Assume that ϕ ∈ W 2,2 (K, ν). Then
(i) |x| |Dϕ| ∈ L 2 (K, ν),
(ii) |Q 1/2 n(y)| |γ(Dϕ)| ∈ L 2 (Σ, µ Σ ),
(iii) the following estimate holds,
∫
∫
|Q 1/2 n(y)| 2 |γ(Dϕ(y))| 2 µ Σ (dy) +
Σ
≤ 2Tr Q
∫
K
∫
|Dϕ(x)| 2 ν(dx) + 4 Tr [Q 2 ]
K
|x| 2 |Dϕ(x)| 2 ν(dx)
K
Tr [(D 2 ϕ(x)) 2 ]|ν(dx).
(2.18)
Proof. Let ϕ ∈ W 2,2 (K, ν) and let {ϕ n } ⊂ C 2 (K) strongly convergent to ϕ
in W 2,2 (K, ν). For i ∈ N we apply (2.14) to D i ϕ n . We have
∫
∫
|Q 1/2 n(y)| 2 |D i ϕ n (y)| 2 µ Σ (dy) + |x| 2 |D i ϕ n (x)| 2 ν(dx)
Σ
≤ 2Tr Q
∫
K
∫
|D i ϕ n (x)| 2 ν(dx) + 4 Tr [Q 2 ]
Summing up on i yields
∫
∫
|Q 1/2 n(y)| 2 |Dϕ n (y)| 2 µ Σ (dy) +
Σ
≤ 2Tr Q
∫
K
K
K
|Dϕ n (x)| 2 ν(dx) + 4 Tr [Q 2 ]
K
|x| 2 |Dϕ n (x)| 2 ν(dx)
∞∑
∫
i,j=1
|DD i ϕ n (x)| 2 ν(dx).
K
|D j D i ϕ n (x)| 2 ν(dx).
Then letting n → ∞ we see that {|Q 1/2 n(y)|γ(Dϕ n )} is strongly convergent
in L 2 (K, ν) and so (i),(ii) and (iii) follow.
When it will be no danger of confusion we shall simply set Dϕ instead of
γ(Dϕ).
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