zt + κF Assumptions We may thus apply the Ito formula see [11] on LF

where ψ = E ∗ r ϕ ∈ W 1,2 B r by Lemma 10.3. So | E r A m h , ϕ | ≤ λ r Q 1 2 r h, hkψk W 1,2 B r ≤ λ r Q 1 2 r h, hkE ∗ r k L W 1,2 ,W 1,2 B r kϕk W 1,2 ≤ c 3 kϕk W d∗,2 . If we define the processes I 1 t = v 0, I 2 t = R t us ds , I 3 t = R t 1 B m f ·, zs, ∇us ds, I 4 t = R t 1 B m g ·, zs, ∇us dW s where the integral I 2 converges in W 1,2 and the integrals I 3 , I 4 converge in L 2 then P [I 2 t ∈ W 2,2 ] = 1, P [vt = I 1 t + A m I 2 t + I 3 t + I 4 t] = 1, t ∈ R + by the Chojnowska-Michalik theorem see [9] or Theorem 13 in [33]. Since E r m ◦ A m can be extended to a linear continuous operator from W 1,2 to W −d ∗ ,2 R by 10.9, E r m vt = E r m I 1 t + E r m A m I 2 t + E r m I 3 t + E r m I 4 t, t ∈ R + in W −d ∗ ,2 R a.s. Since k f ·, zt, ∇utk L 1 B rm∧R ≤ c d R d 2 f ·, utvt + d X i= 1 f i ·, utu x i t L 2 B rm∧R + k f d+ 1 ·, utk L 1 B rm∧R ≤ c d R d 2 1 + 2 1 2 κ 1 2 F 1 2 r m ∧R t, zt + κF r m ∧R t, zt ≤ c R ,d, κ [1 + F r m ∧R t, zt] kg·, zt, ∇utk 2 L 2 B rm∧R ≤ 12κF r m ∧R t, zt 1070 holds for every t ∈ [0, r m ∧ R], we get, using 10.9, kE r m I 1 k C γ [0,r m ∧R],W −d∗,2 R ≤ c 1 kv0k L 2 B rm∧R ≤ c 1 2 1 2 sup t ∈[0,r m ∧R] F 1 2 r m ∧R t, zt, kE r m A m I 2 k C γ [0,r m ∧R],W −d∗,2 R ≤ c 3 1 + R γ sup ≤st≤r m ∧R Q 1 2 r m ∧R R t s ub d b , R t s ub d b t − s γ ≤ c 3 1 + R γ sup ≤st≤r m ∧R R t s Q 1 2 r m ∧R ub, ub d b t − s γ ≤ c 3 1 + R γ R 1 −γ sup t ∈[0,r m ∧R m ] F 1 2 r m ∧R t, zt kE r m I 3 k C γ [0,r m ∧R],W −d∗,2 R ≤ c 2 1 + R γ R 1 −γ sup t ∈[0,r] k f ·, zt, ∇utk L 1 B rm∧R ≤ c R ,d, κ,γ [1 + sup t ∈[0,r m ∧R] F r m ∧R t, zt] E 1 Ω kE r m I 4 k p C γ [0,r m ∧R],W −d∗,2 R ≤ c 1 E § 1 Ω kI 4 k p C γ [0,r m ∧R],L 2 B rm∧R ª ≤ c d ,p, γ,R E 1 Ω Z r m ∧R kg·, z, ∇uk p L 2 H µ ,L 2 B rm∧R ds ≤ c p c d ,p, γ,R E 1 Ω Z r m ∧R kg·, z, ∇uk p L 2 B rm∧R ds ≤ 12κ p 2 R c p c d ,p, γ,R E 1 Ω sup t ∈[0,r m ∧R] F p 2 r m ∧R t, zt where Ω = [F r m ∧R 0, z0 ≤ δ], the estimate of I 4 follows from the Garsia-Rodemich-Rumsey lemma [16], the Burkholder inequality see e.g. [33] and Lemma 3.3. Altogether, E 1 [F rm∧R 0,z0≤δ] kE r m v k p C γ [0,r m ∧R],W −d∗,2 R ≤ c d ,R,r ,c, κ,p,γ,δ by the inequality 10.4. 11 Compactness The present section is the actual core of the paper. We list all preliminary results and assumptions prepared and discussed in previous parts of the paper Section 11.1 and then we carry out, in a few steps, the refined stochastic compactness method as indicated in Section 6. That is to say, we prove tightness of a sequence of solutions of approximating equations Section 11.2, then we verify the assumptions of the Jakubowski-Skorokhod theorem A.1 and prove that the limit process yielded by this theorem is the solution of 1.1 claimed in Theorem 5.1 Sections 11.3 - 11.8, whereas the proof of Theorem 5.2 is given simultaneously in Section 11.6. 1071

11.1 Assumptions

Let i µ be a finite spectral measure on R d , ii Θ be a Borel probability measure on H l oc , iii for every m ∈ N and i ∈ {0, . . . , d}, f i m , g i m : R d × R n → R n ×n and f d+ 1 m , g d+ 1 m : R d × R n → R n be measurable functions satisfying the assumptions of Lemma 9.1, iv for every m ∈ N, F m : R d × R n → R + be measurable functions satisfying the assumptions a-c of Proposition 8.1, v κ ∈ R + be such that 10.1-10.3 and 10.8 hold for f i m , g i m : i ∈ {0, . . . , d + 1}, F m and y ∈ R n for a.e. w ∈ B m , for every m ∈ N, vi for i ∈ {0, . . . , d}, f i , g i : R d × R n → R n ×n , f d+ 1 , g d+ 1 : R d × R n → R n , F : R d × R n → R + be measurable functions such that, for a.e. w ∈ R d , there is f i m w, · → f i w, ·, g i m w, · → g i w, ·, F m w, · → Fw, · uniformly on compact sets in R n for every i ∈ {0, . . . , d + 1}, vii with the notation 4.1, 4.1, z m = u m , v m be H -continuous F m t -adapted solutions of u t t = ∆u + 1 B m f m ·, z, ∇u + 1 B m g m ·, z, ∇u dW m on completely filtered stochastic bases Ω m , F m , F m t , P m for some spatially homogeneous F m t -Wiener processes W m with spectral measure µ, such that z m 0 is supported on some ball in H with a radius dependent on m for every m ∈ N see Lemma 9.1 and lim m →∞ P m π R z m 0 ∈ · − Θ π R ∈ · = 0 11.1 holds for every R 0 where the norm is taken in the total variation of measures on BH R and π R : H → H R is the restriction operator see Section 2, viii it hold that Θ {u, v ∈ H l oc : kF ∗ ·, uk L 1 B R ∞} = 1, R 11.2 where F ∗ = sup F m , ix given r 0, E r be an extension operator on L 2 B r and on W 1,2 B r , i.e. E r : L 2 B r → L 2 R d , E r : W 1,2 B r → W 1,2 R d 11.3 are linear continuous operators such that E r h = h a.e. on B r , 1072 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.