holds also for h = 1 Martingale property

holds for every k ∈ N, R ∈ [0, r m j ] and t ∈ [0, T λ] by 11.10. Applying Fatou’s lemma on the LHS of 11.12 and the Lebesgue dominated convergence theorem on the RHS of 11.12, we obtain, letting j → ∞, E ¨ h E R π R z0 sup r ∈[0,t] L k Gr, zr « ≤ 4e ρ t Z H l oc h E R π R zL k G0, z dΘ for every t ∈ [0, T λ] and m ∈ N by 11.1, 11.9 and ii, iii in Section 11.3 as L k is a bounded nondecreasing continuous and eventually constant function. Since E R ◦ π R : H l oc → H l oc converges uniformly to identity on H l oc as R → ∞, we get E ¨ h z0 sup r ∈[0,t] L k Gr, zr « ≤ 4e ρ t Z H l oc hzL k G0, z dΘ 11.13 for every t ∈ [0, T λ] and k ∈ N by the Lebesgue dominated convergence theorem. Consequently,

11.13 holds also for h = 1

K where K is closed in H l oc , whence also for every F σ -set and every Borel set K ⊆ H l oc by regularity of Θ = P [z0 ∈ ·] Remark 11.1. The claim now follows from Fatou’s lemma when letting k → ∞, applied on the LHS, since L k ≤ L for every k ∈ N, applied on the RHS.

11.7 Martingale property

Let us remind the reader that the integrals in the following Proposition converge by the assumption v in Section 11.1 and by 11.11. Proposition 11.9. Let ϕ ∈ D. Then 〈vt, ϕ〉 = 〈v0, ϕ〉 + Z t 〈ur, A ϕ〉 d r + Z t 〈 f ·, ur, vr, ∇ur, ϕ〉 d r + Z t 〈g·, ur, vr, ∇ur dW r , ϕ〉 holds a.s. for every t ≥ 0 where W was defined in Corollary 11.6. Proof. Let k ∈ N, let ϕ ∈ D have support in B k and, throughout this proof, consider only j ∈ N such that r m j ≥ T k , i.e. j ≥ j for some j and it holds that k ≤ T k ≤ r m j ≤ T r m j ≤ m j , j ≥ j . Fixing 0 ≤ s t ≤ k, we consider the sequence ϕ i from Corollary C.1. Let also J ∈ N, 0 ≤ s 1 ≤ 1079 · · · ≤ s J ≤ s, let H : R 2 J ×J × R dim H µ J × R N + → [0, 1] be a continuous function and define X 1 j = E r m j u m j s i ∧ r m j E r m j v m j s i ∧ r m j , ϕ i 1 + L 2 i ,i 1 ≤J X 2 j = W m j s 1 e l l , . . . , W m j s J e l l , ‚ F m j ·, E r m j u m j L 1 B Tρ Œ ρ∈N X j = X 1 j , X 2 j X j = € 〈z j s i , ϕ i 1 〉 L 2 Š i ,i 1 ≤J , β j s 1 , . . . , β j s J , ‚ F m j ·, u j L 1 B Tρ Œ ρ∈N X = € 〈zs i , ϕ i 1 〉 L 2 Š i ,i 1 ≤J , βs 1 , . . . , βs J , ν for j ≥ j . If h δ : R + → [0, 1] 11.14 is any continuous function with support in [0, δ] such that h δ = 1 on [0, δ2] then we also define continuous mappings d j q : CR + ; H → R u, v 7→ h δ ˜ F m j 0, u0, v0 〈vq, ϕ〉 − 〈v0, ϕ〉 − h δ ˜ F m j 0, u0, v0 Z q 〈ur, A ϕ〉 d r − h δ ˜ F m j 0, u0, v0 Z q 〈 f m j ·, ur, vr, ∇ur, ϕ〉 d r D j ,l q : CR + ; H → R u, v 7→ h δ ˜ F m j 0, u0, v0 Z q 〈g m j ·, ur, vr, ∇ure l , ϕ〉 d r D j q : CR + ; H → R u, v 7→ h 2 δ ˜ F m j 0, u0, v0 X l Z q 〈g m j ·, ur, vr, ∇ure l , ϕ〉 2 d r for q ∈ [0, k], j ≥ j and l indexing the ONB e l in H µ that satisfy |d j q z| + |D j ,l q z| + |D j q z| ≤ K1 [˜ F m j 0,z0≤δ] [1 + sup r ∈[0,k] ˜ F m j r, zr] 11.15 for q ∈ [0, k], z ∈ CR + ; H , j ≥ j , l up to dim H µ and for some K = K d ,k, κ,a,ϕ,c as k f m j ·, ur, vr, ∇urk L 1 B k ≤ 8κ 1 2 Leb d B k + κ[1 + ˜ F m j r, zr] 11.16 kg m j ·, ur, vr, ∇urk L 2 B k ≤ 5κ 1 2 ˜ F 1 2 m j r, zr 1080 holds for every r ∈ [0, k] where ˜F m = ˜ F m λ Tk ,0,T k is the conic energy function for ˜ F m w, y = F m w, y + | y| 2 2 defined as in 2.2. Also, for every p 0, there exist constants K p depending also on d, k, κ, a, ϕ and c such that E sup q ∈[0,k] h |d j q z

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