Tightness We may thus apply the Ito formula see [11] on LF

x us define the processes Z m t = U m t, V m t := E r m u m t ∧ r m , E r m v m t ∧ r m , t ∈ R + where r m and E r m were defined in 10.6 and 10.7, xi it hold that Z B R sup {F ∗ w, y : | y| ≤ R} dw ∞, R 0, 11.4 xii given r 0, there exist α R = α r ,R for R 0 such that lim R →∞ α R = 0 and | f d+ 1 m w, y| ≤ α R F m w, y, | y| ≥ R 11.5 holds for every m ∈ N and almost every w ∈ B r . Remark 11.1. Observe that 11.1 implies P m [Z m 0 ∈ ·] → Θ weakly on H l oc . Indeed, let E r : H r → H be a continuous linear extension operator, i.e. E r z = z a.e. on B r , and let ϕ : H l oc → [0, 1] be a uniformly continuous function. Then, for every ǫ 0, there is r 0 such that |ϕz − ϕE r π r z| ≤ ǫ, z ∈ H l oc . Thus lim m →∞ Z H l oc ϕZ m 0 dP m = Z H l oc ϕ dΘ and the claim follows from Theorem 2.1 in [1]. Remark 11.2. Observe also that, given R 0, the real valued sequence kF m ·, U m 0k L 1 B R is tight in R + . Indeed, there is P m ” kF m ·, U m 0k L 1 B R δ — = P m ” kF m ·, π R u m 0k L 1 B R δ — ≤ kP m π R z m 0 ∈ · − Θ π R ∈ · k + Θ ¦ u, v : kF m ·, uk L 1 B R δ © ≤ ǫ R ,m + Θ ¦ u, v : kF ∗ ·, uk L 1 B R δ © for m ∈ N such that r m ≥ R where ǫ R ,m → 0 by 11.1. Tightness follows from 11.2.

11.2 Tightness

Lemma 11.3. The sequence of processes Z m is tight in Z = C w R + , W 1,2 l oc × C w R + , L 2 l oc . Proof. Let ǫ ∈ 0, 1, let us define ˜ F m w, y = F m w, y + | y| 2 2, ˜ F ∗ w, y = F ∗ w, y + | y| 2 2 and consider their conic energy functions ˜ F m ,k = ˜ F m λ Tk∧rm ,0,T k ∧rm , ˜ F ∗ k = ˜ F ∗ 0,0,T k 1073 for k ∈ N defined as in 2.2 with the notation 2.3 and 10.6. Let also p ∈ 1, ∞ and γ ∈ 0, 1 satisfy γ + 2 p 1 2 and let d ∗ = ” d 2 — + 1. Since |g d+ 1 m w, y| 2 + |∇ y ˜ F m w, y + g d+ 1 m w, y| 2 = |g d+ 1 m w, y| 2 + |∇ y F m w, y + y + g d+ 1 m w, y| 2 ≤ 2|g d+ 1 m w, y| 2 + 2|∇ y F m w, y + g d+ 1 m w, y| 2 + 2| y| 2 ≤ 2κ + 4 ˜ F m w, y the assumptions 5.4-10.3, 10.8 are satisfied for κ, ˜ F m for every y ∈ R n and a.e. w ∈ B m for the constant ˜k which depends on κ and p, and so Lemma 10.1 and Lemma 10.4 applied on ˜ F m and Lx = x p 2 yield, for every δ 0, Z [˜ F m ,k

0,z

m 0≤δ] sup t ∈[0,k∧r m ] ˜ F p 2 m ,k t, z m t dP m ≤ 4e ρk Z [˜ F m ,k

0,z

m 0≤δ] ˜ F p 2 m ,k

0, z

m 0 dP m ≤ 4e ρk δ p 2 where ρ = ρ c, κ,p so Z [˜ F m ,k

0,z

m 0≤δ] ¨ sup t ∈[0,k] kU m tk p W 1,2 B k + sup t ∈[0,k] kV m tk p L 2 B k « dP m ≤ C k , δ 11.6 as sup t ∈[0,k] kU m tk W 1,2 B k ≤ max {1, ˜ α k } sup t ∈[0,k∧r m ] ku m tk W 1,2 B k ∧rm , sup t ∈[0,k] kV m tk L 2 B k ≤ max {1, ˜ α k } sup t ∈[0,k∧r m ] kv m tk L 2 B k ∧rm , ku m tk 2 W 1,2 B k ∧rm + kv m tk 2 L 2 B k ∧rm ≤ 2 max {α k , 1 }˜F m ,k t, z m t, t ∈ [0, k ∧ r m ] where α k = sup w ∈B Tk ka −1 wk, ˜ α k = max {kE r m k L L 2 B rm ,L 2 R d , kE r m k L W 1,2 B rm ,W 1,2 R d : m ∈ N, r m ≤ k}, and Z [˜ F m ,k

0,z

m 0≤δ] kU m k p C γ [0,k],L 2 B k + kV m k p C γ [0,k],W −d∗,2 k dP m ≤ C k , δ 11.7 by Lemma 10.4 for some C k , δ ∈ R + depending also on c, a, p, d, γ, r j j ∈N , E r j j ∈N , E r j j ∈N and κ since kU m k C γ [0,k];L 2 B k ≤ max {1, ˜ β k }  ku m 0k L 2 B k ∧rm + 2k sup t ∈[0,k∧r m ] kv m tk L 2 B k ∧rm   where ˜ β k = max {kE r m k L L 2 B rm ,L 2 R d : m ∈ N, r m ≤ k}. 1074 Since P m ” ˜ F m ,k

0, z

m 0 δ — = P m ” ˜ F m ,k 0, π T k z m 0 δ — ≤ Θ ” ˜ F m ,k 0, π T k δ — + sup A BH Tk P m ” π T k z m 0 ∈ A — − Θ ” π T k ∈ A — = ǫ m ,k + Θ ” ˜ F m ,k 0, · δ — ≤ ǫ m ,k + Θ ” ˜ F ∗ k 0, · δ — holds for every m, k ∈ N and δ where the norms are in the total variation of measures on H T k , taking 11.1 and 11.2 into account, we can find δ k 0 and a k 0 so that P m ” ˜ F m ,k

0, z

m 0 δ k — ≤ ǫ 3 · 4 k , a k ≥   6 · 4 k · 2 p · C k , δ k ǫ   1 p holds for every m, k ∈ N. Then P m – sup t ∈[0,k] kU m k W 1,2 B k + kU m k C γ [0,k];L 2 B k a k ™ ≤ P m ” ˜ F m ,k

0, z

m 0 δ k — + P m – ˜ F m ,k

0, z

m 0 ≤ δ k , sup t ∈[0,k] kU m k W 1,2 B k a k 2 ™ + P m • ˜ F m ,k

0, z

m 0 ≤ δ k , kU m k C γ [0,k];L 2 B k a k 2 ˜ ≤ ǫ 3 · 4 k + 2 p a p k Z [˜ F m ,k

0,z

m 0≤δ k ] ¨ sup t ∈[0,k] kU m k p W 1,2 B k + kU m k p C γ [0,k];L 2 B k « dP m ≤ ǫ 4 k by 11.6 and 11.7, and analogously P m – sup t ∈[0,k] kV m k L 2 B k + kV m k C γ [0,k];W −d∗,2 k a k ™ ≤ ǫ 4 k . If K 1 = ¦ h ∈ C w R + ; W 1,2 l oc : khk L ∞ 0,k;W 1,2 B k + khk C γ [0,k];L 2 B k ≤ a k , k ∈ N © K 2 = § h ∈ C w R + ; L 2 l oc : khk L ∞ 0,k;L 2 B k + khk C γ [0,k];W −d∗,2 k ≤ a k , k ∈ N ª then K 1 × K 2 is compact in Z by Corollary B.1 and P m Z m ∈ K 1 × K 2 1 − ǫ, m ∈ N. 1075

11.3 Skorokhod representation