The Limit along a Subsequence

exists and is measurable. Furthermore, for all ω ∈ Ω , ˆ η t t ∈[0,T ]\C satisfies ˆ η t = v + Z t D bX r x · ˆ η k r d r, t ∈ [0, inf C, ˆ η t = ˆ η rt + Z t rt D bX r x · ˆ η k r d r, t ∈ [inf C, T ]\C. 3.8 We stress here that the typical ω is fixed at the beginning, in particular before considering any sequences or subsequences. At the beginning of the proof of Proposition 3.10 we will choose the set Ω with full measure. No sequence or subsequence is involved in this choice. The statement of the Proposition will then follow by completely pathwise arguments, which are purely deterministic and do not require any other choice of events. In the next step we extend ˆ ηω, ω ∈ Ω , to a right-continuous trajectory on [0, T ]. Then, we prove that in coordinates of the moving frame introduced in Section 2.2 ˆ η solves the evolution equation appearing in Theorem 2.5, which is shown to admit a pathwise unique solution. Finally, we outline how the almost sure directional differentiability can be deduced from this. First note that η T ǫ converges a.s. if and only if for every component η i T ǫ, i = 1, . . . , d, lim inf ǫ→0 η i T ǫ = lim sup ǫ→0 η i T ǫ a.s. 3.9 Let now ǫ i, − ν ν and ǫ i,+ ν ν be two random sequences, along which η i T ǫ converges to its limes inferior and its limes superior, respectively. We apply Proposition 3.10 to these two sequences and get two limiting objects ˆ η − and ˆ η + , being two trajectories in R d whose i-th components at time T , ˆ η i, − T and ˆ η i,+ T , coincide with the limes inferior and limes inferior of η i T ǫ, of course. From the fact that both ˆ η − and ˆ η + solve an equation having a pathwise unique solution we conclude that P h ˆ η − t = ˆ η + t , ∀t ∈ [0, T ] i = 1, which implies ˆ η i, − T = ˆ η i,+ T a.s. and we obtain 3.9.

3.3.1 The Limit along a Subsequence

Proof of Proposition 3.10. Let T 0 still be fixed. We choose Ω ⊆ Ω with full measure such that for all ω ∈ Ω Lemma 3.1 holds and Corollary 3.9 holds for all n. Let now ω ∈ Ω be fixed. Let t ∈ [0, T ]\C and n be such that t ∈ A n . Using Proposition 3.2 there exists ∆ n = ∆ n ω 0 such that l q n x = l q n x ǫ = 0 if q n inf C and both of them are strictly positive if q n inf C for all |ǫ| ∈ 0, ∆ n . Then, η t ǫ =η rq n ǫ + 1 ǫ Z t rq n bX r x ǫ − bX r x d r + 1 ǫ Z r ǫ q n nX r x ǫ d l r x ǫ − 1 ǫ Z rq n nX r x ǫ d l r x ǫ =η rq n ǫ + d X k=1 Z t rq n   Z 1 ∂ b ∂ x k X α,ǫ r dα   η k r ǫ d r + R q n x ǫ , 3.10 863 where X α,ǫ r := αX r x ǫ + 1 − αX r x, α ∈ [0, 1], and R q n x ǫ := 1 ǫ Z r ǫ q n rq n nX r x ǫ d l r x ǫ . 3.11 Note that if q n inf C we have rq n = 0, η rq n ǫ = v and R q n x ǫ = 0. In any case, kR q n x ǫ k ≤ 1 ǫ Z r ǫ q n rq n knX r x ǫ k dl s x ǫ = l r ǫ q n x ǫ − l rq n x ǫ ǫ −→ 0 as ǫ → 0, 3.12 by Corollary 3.9. Recall that kη t ǫk ≤ expc 1 T + l T x + l T x ǫ ≤ c for all t ∈ [0, T ] and ǫ 6= 0 by Proposition 3.2 and Lemma 3.1. Let ǫ ν ν = ǫ ν ω ν be any random sequence converging to zero. By a diagonal procedure, we can extract a subsequence ǫ ν l l with ν l l = ν l ω l such that η rq n ǫ ν l has a limit ˆ η rq n ∈ R d as l → ∞ for all n ∈ N. Let now ˆ η : [0, T ]\C → R d be the unique solution of ˆ η t := ˆ η rq n + Z t rq n D bX r x · ˆ η r d r, t ∈ A n . By 3.10 and Proposition 3.2, we get for |ǫ| ∈ 0, ∆ n and t ∈ A n , kη t ǫ − ˆ η t k ≤kη rq n ǫ − ˆ η rq n k + kR q n x ǫ k + sup r ∈A n D bX α,ǫ r − DbX r x expc 1 T + l T x + l T x ǫ + c 2 Z t kη r ǫ − ˆ η r k d r. Since η rq n ǫ ν l → ˆ η rq n , kR q n x ǫ k → 0 , X α,ǫ νl r → X r x uniformly in r ∈ [0, t] and since the derivatives of b are continuous, we obtain by Gronwall’s Lemma that η t ǫ ν l converges to ˆ η t uni- formly in t ∈ A n for every n. Thus, since C has zero Lebesgue measure, η t ǫ ν l converges to ˆ η t for all t ∈ [0, T ]\C as l → ∞ and by the dominated convergence theorem we get that ˆ η t ω t ∈[0,T ]\C satisfies 3.8. Finally, the measurability of ˆ η is immediate from its construction. From now on we will denote by ˆ η the limiting object constructed in Proposition 3.10 from any arbitrary but fixed random sequence ǫ n . The next lemma shows that ˆ η rq n is in the tangent at X rq n x for every q n . Lemma 3.11. For every q n inf C, i 〈η rq n ǫ, nX rq n x〉 → 0 a.s. and in L p , p 1, as ǫ → 0, ii 〈η r ǫ q n ǫ, nX r ǫ q n x ǫ 〉 → 0 a.s. and in L p , p 1, as ǫ → 0. Proof. By dominated convergence it suffices to prove convergence almost surely. Let ℓ be such that q n ∈ [τ ℓ , τ ℓ+1 . Then, clearly X rq n x ∈ U m ℓ ∩ ∂ G. Recall that nX rq n x = ∇u 1 m ℓ X rq n x, and by Taylor’s formula we get 〈η rq n ǫ, nX rq n x〉 = 1 ǫ u 1 m ℓ X rq n x ǫ − u 1 m ℓ X rq n x + Oǫ. 864 Note that the term of second order in the Taylor expansion is in O ǫ by Proposition 3.2. Recall that u 1 m ℓ X rq n x = 0, and combining formula 2.4 and 3.4, we get u 1 m ℓ X rq n x ǫ = u 1 m ℓ X τ ℓ x ǫ + M x, ℓ rq n ǫ + L rq n x ǫ = M x, ℓ rq n ǫ − M x, ℓ r ǫ rq n ǫ for all ǫ ∈ −∆ n , ∆ n for some positive ∆ n . Arguing similarly as in 3.7 we obtain from Corollary 3.9 i that u 1 m ℓ X rq n x ǫ − u 1 m ℓ X rq n x ǫ ≤ 2 M x, ℓ rq n ǫ − M x, ℓ r ǫ q n ǫ ǫ −→ 0 as ǫ → 0, and i follows. The proof of ii is rather analogous. For an appropriate ∆ n 0 we have r ǫ q n ∈ [τ ℓ , τ ℓ+1 and l r ǫ q n x 0 for all |ǫ| ∈ 0, ∆ n . Then, for such ǫ we get again by using Taylor’s formula and the fact that u 1 m ℓ X r ǫ q n x ǫ = 0, 〈η r ǫ q n ǫ, nX r ǫ q n x ǫ 〉 =〈η r ǫ q n ǫ, ∇u 1 m ℓ X r ǫ q n x ǫ 〉 = − 1 ǫ u 1 m ℓ X r ǫ q n x ǫ − u 1 m ℓ X r ǫ q n x + Oǫ = 1 ǫ M x, ℓ r ǫ q n − M x, ℓ rr ǫ q n + Oǫ. Since M x, ℓ attains its minimum over [ τ ℓ , q n ] at time rq n and its minimum over [τ ℓ , r ǫ q n ] at time rr ǫ q n , respectively, we finally get |〈η r ǫ q n ǫ, nX r ǫ q n x ǫ 〉| ≤ 1 |ǫ| M x, ℓ r ǫ q n − M x, ℓ rq n + M x, ℓ rr ǫ q n − M x, ℓ rq n + Oǫ ≤ 2 |ǫ| M x, ℓ r ǫ q n − M x, ℓ rq n + Oǫ, which tends to zero again by Corollary 3.9 i. So far ˆ η t is not defined for every t ∈ C, only for the left endpoints rq n of the excursion intervals. We will now extend the trajectories of ˆ η to the set C. To that aim, note that since for every m ≥ 0 the coordinate mapping u m is a diffeomorphism, the set {∇u i m x, i = 2, . . . , d} is linear independent for all x ∈ U m and by construction it is also a basis of the tangent space at x if x ∈ ∂ G ∩ U m . Let {¯n m 2 x, . . . , ¯ n m d x} be the Gram-Schmidt orthonormalization of {∇u i m x, i = 2, . . . , d} for every x ∈ U m and for every m. Then, ¯ n m x := {nx, ¯n m 2 x, . . . , ¯ n m d x} is an ONB of R d for all x ∈ U m ∩ ∂ G. We define now ˆ η t for t ∈ C ∩ [τ ℓ , τ ℓ+1 in the coordinates w.r.t. the basis n m ℓ X t x on U m ℓ ∩ ∂ G. For that purpose it is sufficient to define 〈 ˆ η t , ∇u i m ℓ X t x〉 for i ∈ {1, . . . , d}. We set η ∗ t := ˆ η t if t ∈ [0, T ]\C, if t ∈ [0, T ] ∩ C and for i = 1, . . . , d we define the process I ∗ i t t ∈[0,T ] on [0, T ] by I ∗ i t := ∇u i m ℓ X τ ℓ x· ˆ η τ ℓ + Z t τ ℓ ∇b i m ℓ X r x·η ∗ r d r+ d X j=1 Z t τ ℓ ∇σ i j m ℓ X r x·η ∗ r d w j r , if t ∈ [τ ℓ , τ ℓ+1 . 865 Now we define for t ∈ C ∩ [τ ℓ , τ ℓ+1 〈 ˆ η t , ∇u 1 m ℓ X t x〉 =〈 ˆ η t , nX t x〉 := 0 and 〈 ˆ η t , ∇u i m ℓ X t x〉 := I ∗ i t for i = 2, . . . , d. Having extended ˆ η to a trajectory on [0, T ] we can define ˆI i t, t ∈ [0, T ], similarly to I ∗ i t locally on each interval [τ ℓ , τ ℓ+1 . Note that I ∗ i and ˆI i are continuous on each interval [τ ℓ , τ ℓ+1 , in particular they are right-continuous on [0, T ], and since C has zero Lebesgue measure, I ∗ i t = ˆI i t a.s. for every t. Thus, by a continuity argument P h I ∗ i = ˆ I i on [0, T ] i = 1. 3.13 Lemma 3.12. For every ℓ ≥ 0, i = 1, . . . , d and p ≥ 2 we have E   Z τ ℓ+1 τ ℓ Z 1 ∇b i m ℓ X α,ǫ νk r dα · η r ǫ ν k − ∇b i m ℓ X r x · ˆ η r p d r   → 0 as k → ∞, and E   sup s ∈[τ ℓ , τ ℓ+1 Z s τ ℓ Z 1 ∇σ i j m ℓ X α,ǫ νk r dα · η r ǫ ν k − ∇σ i j m ℓ X r x · ˆ η r d w j r p   → 0 as k → ∞, for every j = 1, . . . , d, where as before X α,ǫ r := αX r x ǫ + 1 − αX r x, α ∈ [0, 1]. Proof. By Proposition 3.2 and Lemma 3.1 the first term can be estimated as follows E   Z τ ℓ+1 τ ℓ Z 1 ∇b i m ℓ X α,ǫ r dα · η r ǫ − ∇b i m ℓ X r x · ˆ η r p d r   ≤c 1 E   Z τ ℓ+1 τ ℓ Z 1 ∇b i m ℓ X α,ǫ r dα − ∇b i m ℓ X r x p d r   + c 2 E   Z τ ℓ+1 τ ℓ kη r ǫ − ˆ η r k p d r   . For the second term we get similarly, using Burkholder’s inequality, E   sup s ∈[τ ℓ , τ ℓ+1 Z s τ ℓ Z 1 ∇σ i j m ℓ X α,ǫ r dα · η r ǫ − ∇σ i j m ℓ X r x · ˆ η r d w j r p   ≤ c 3 E   Z τ ℓ+1 τ ℓ Z 1 ∇σ i j m ℓ X α,ǫ r dα · η r ǫ − ∇σ i j m ℓ X r x · ˆ η r p d r   ≤ c 4 E   Z τ ℓ+1 τ ℓ Z 1 ∇σ i j m ℓ X α,ǫ r dα − ∇σ i j m ℓ X r x p d r   + c 5 E   Z τ ℓ+1 τ ℓ kη r ǫ − ˆ η r k p d r   . Hence both terms converges to zero along ǫ ν k by dominated convergence, since X α,ǫ νk r → X r x uniformly in r ∈ [0, T ], ∇b i m ℓ and ∇σ i j m ℓ are continuous and η r ǫ ν k converges to ˆ η r uniformly in r ∈ A n for every n. For every i = 1. . . . , d let F i t t ∈[0,T ] be the process defined by F i t := 〈 ˆ η t , ∇u i m ℓ X t x〉 if t ∈ [τ ℓ , τ ℓ+1 . 866 Lemma 3.13. For almost every ω the following holds. i For all t ∈ [0, T ] F 1 t = I t := ˆI 1 t if t inf C ℓ , ˆ I 1 t − ˆI 1 rt if t ≥ inf C ℓ , with ℓ such that t ∈ [τ ℓ , τ ℓ+1 . ii For every i ∈ {2, . . . , d} we have for all t ∈ [0, T ] F i t = ˆI i t =∇u i m ℓ X τ ℓ x · ˆ η τ ℓ + Z t τ ℓ ∇b i m ℓ X r x · ˆ η r d r + d X j=1 Z t τ ℓ ∇σ i j m ℓ X r x · ˆ η r d w j r , 3.14 with ℓ such that t ∈ [τ ℓ , τ ℓ+1 . In particular, the trajectories of ˆ η are right-continuous on [0, T ]. Proof. i First note that since ˆ I 1 is continuous on each interval [ τ ℓ , τ ℓ+1 and t 7→ rt is right- continuous, the paths of I are also right-continuous on [0, T ]. Let t ∈ [0, T ] be fixed, ℓ be such that t ∈ [τ ℓ , τ ℓ+1 and ∆ T 0 be as in Remark 3.3. Then, t 6∈ C a.s. Further, if t inf C ℓ and |ǫ| ∆ T then l t x = l τ ℓ x and l t x ǫ = l τ ℓ x ǫ . So we have by Taylor’s formula and 2.2 that F 1 t =〈 ˆ η t , ∇u 1 m ℓ X t x〉 = lim k →∞ 1 ǫ ν k u 1 m ℓ X t x ǫ νk − u 1 m ℓ X t x 1l {0|ǫ νk |∆ T } = lim k →∞ 1 ǫ ν k u 1 m ℓ X τ ℓ x ǫ νk − u 1 m ℓ X τ ℓ x + Z t τ ℓ b 1 m ℓ X r x ǫ νk − b 1 m ℓ X r x d r + d X j=1 Z t τ ℓ σ 1 j m ℓ X r x ǫ νk − σ 1 j m ℓ X r x d w j r    = lim k →∞ ∇u 1 m ℓ X τ ℓ x · η τ ℓ ǫ ν k + Z t τ ℓ Z 1 ∇b 1 m ℓ X α,ǫ νk r · η r ǫ ν k dα d r + d X j=1 Z t τ ℓ Z 1 ∇σ 1 j m ℓ X α,ǫ νk r · η r ǫ ν k dα d w j r    , where as before X α,ǫ r := αX r x ǫ + 1 − αX r x, α ∈ [0, 1]. On the other hand, if t inf C ℓ we get F 1 t =〈 ˆ η t , ∇u 1 m ℓ X t x〉 = lim k →∞ ∇u 1 m ℓ X rt x · η rt ǫ ν k + Z t rt Z 1 ∇b 1 m ℓ X α,ǫ νk r · η r ǫ ν k dα d r + d X j=1 Z t rt Z 1 ∇σ 1 j m ℓ X α,ǫ νk r · η r ǫ ν k dα d w j r + 1 ǫ ν k  l r ǫνk t x ǫ νk − l rt x ǫ νk ‹    , 867 where the first term and the last term converge to zero by Corollary 3.9ii and Lemma 3.11, re- spectively. The remaining terms converge in L 2 to the corresponding terms in the definition of ˆI 1 by Lemma 3.12. Hence, we obtain that F 1 t = I t a.s. for every t. Since the trajectories of ˆ η are continuous on every excursion interval A n , the paths of F 1 are right-continuous on every A n and have only possibly jumps at the step times τ ℓ . Hence, P h F 1 t = I on [0, T ]\C i = P h 〈 ˆ η . , ∇u i m ℓ X . x〉 = I on A n for all n i = 1. Finally, since by definition F 1 = 〈 ˆ η . , ∇u i m ℓ X . x〉 = 0 = I on C ∩ [τ ℓ , τ ℓ+1 for every ℓ, we obtain P h F 1 = I on [0, T ] i = 1, which gives i. ii We proceed similarly to i. Let i ∈ {2, . . . , d} and t ∈ [0, T ] be fixed and ℓ be such that t ∈ [τ ℓ , τ ℓ+1 . Then, t 6∈ C a.s. and we have by Taylor’s formula and 2.2 F i t =〈 ˆ η t , ∇u i m ℓ X t x〉 = lim k →∞ ∇u i m ℓ X τ ℓ x · η τ ℓ ǫ ν k + Z t τ ℓ Z 1 ∇b i m ℓ X α,ǫ νk r · η r ǫ ν k dα d r + d X j=1 Z t τ ℓ Z 1 ∇σ i j m ℓ X α,ǫ νk r · η r ǫ ν k dα d w j r    , The sequence on the right hand side converges in L 2 to the right hand side of 3.14 by Lemma 3.12. Hence, we obtain that F i t = ˆI i t = I ∗ i t a.s. for every t. Since the trajectories of ˆ η are continuous on every excursion interval A n , the path of F i are right-continuous on every A n and we get P ” F i = ˆ I i = I ∗ i on [0, T ] \C — = P ” F i = ˆ I i = I ∗ i on A n for all n — = 1. Finally, since by definition F i = 〈 ˆ η . , ∇u i m ℓ X . x〉 = I ∗ i on C ∩ [τ ℓ , τ ℓ+1 , we use 3.13 to obtain P ” F i = ˆ I i = I ∗ i on [0, T ] — = 1, and we obtain ii. The right-continuity of the trajectories of ˆ η is now immediate from i and ii. Indeed, writing ˆ η t in the basis n m ℓ X t x, we get that on one hand the basis vectors are continuous in t on [τ ℓ , τ ℓ+1 and on the other hand the coordinates are right-continuous in t by i and ii. The extension of ˆ η on C and Lemma 3.11 imply that 〈 ˆ η t , ∇u 1 m ℓ X t x〉 = 〈 ˆ η t , nX t x〉 = 0 for all t ∈ [τ ℓ , τ ℓ+1 ∩ C, i.e. when the process X x is at the boundary ˆ η is at the tangent space, while the projection of ˆ η is a continuous process as indicated by 3.14. Let now for all x ∈ U m , m ≥ 0 and η ∈ R d ˜ Π m x η := d X k=2 〈η, ¯n m k x〉 ¯n m k x, 3.15 868 so that obviously ˜ Π m x η = π x η, ∀x ∈ ∂ G ∩ U m , ∀η ∈ R d . For later use we prove now uniform convergence of ˜ Π m ℓ X t x ǫ η t ǫ to ˜ Π m ℓ X t x ˆ η t along the chosen subsequence. The proof is based on the fact that there are no local time terms appearing in equation 2.2 for u i m ℓ , i = 2, . . . , d. In particular, note that ˜ Π m ℓ X t x ˆ η t is not the same as Q · O t · ˆ η t . Later we will identify that process with Y 2 t appearing in Theorem 2.5, which does depend on the local time. Both processes do only coincide for t ∈ [τ ℓ , τ ℓ+1 ∩ C. Lemma 3.14. Let ∆ T 0 be as in Remark 3.3 such that, for every ℓ ≥ 0, ˜ Π m ℓ X s x ǫ η s ǫ is well defined for all s ∈ [τ ℓ , τ ℓ+1 and all 0 |ǫ| ∆ T . Then, sup s ∈[τ ℓ , τ ℓ+1 ˜ Π m ℓ X s x ǫνk η s ǫ ν k − ˜ Π m ℓ X s x ˆ η s 1l {0|ǫ νk |∆ T } → 0 in L p , p ≥ 2 as k → ∞. Proof. Since every function ¯ n m k is continuous on U m , it suffices by Proposition 3.2 to show that sup s ∈[τ ℓ , τ ℓ+1 ˜ Π m ℓ X s x η s ǫ ν k − ˜ Π m ℓ X s x ˆ η s 1l {0|ǫ νk |∆ T } → 0 in L p , p ≥ 2, as k → ∞, and for this it is enough to prove that for every i ∈ {2, . . . , d}, sup s ∈[τ ℓ , τ ℓ+1 〈η s ǫ ν k , ∇u i m ℓ X s x〉 − 〈 ˆ η s , ∇u i m ℓ X s x〉 1l {0|ǫ νk |∆ T } → 0 in L p as k → ∞. For |ǫ| ∆ T we use as before Taylor’s formula and 2.2 to obtain 〈η s ǫ, ∇u i m ℓ X s x〉 = 1 ǫ u i m ℓ X s x ǫ − u i m ℓ X s x + Oǫ =∇u i m ℓ X τ ℓ x · η τ ℓ ǫ + Z s τ ℓ Z 1 ∇b i m ℓ X α,ǫ r · η r ǫ dα d r + d X j=1 Z s τ ℓ Z 1 ∇σ i j m ℓ X α,ǫ r · η r ǫ dα d w j r + Oǫ, 3.16 where again X α,ǫ r := αX r x ǫ + 1 − αX r x, α ∈ [0, 1]. Comparing 3.14 and 3.16 leads to E   sup s ∈[τ ℓ , τ ℓ+1 〈η s ǫ, ∇u i m ℓ X s x〉 − 〈 ˆ η s , ∇u i m ℓ X s x〉 p 1l {0|ǫ|∆ T }   ≤c 1 E h k∇u i m ℓ X τ ℓ xk p kη τ ℓ ǫ − ˆ η τ ℓ k p i + c 1 E   Z τ ℓ+1 τ ℓ Z 1 ∇b i m ℓ X α,ǫ r dα · η r ǫ − ∇b i m ℓ X r x · ˆ η r p d r   + c 1 d X j=1 E   sup s ∈[τ ℓ , τ ℓ+1 Z s τ ℓ Z 1 ∇σ i j m ℓ X α,ǫ r dα · η r ǫ − ∇σ i j m ℓ X r x · ˆ η r d w j r p   + Oǫ. The claim follows now from Lemma 3.12 and the fact that η τ ℓ ǫ ν k → ˆ η τ ℓ . 869

3.3.2 A Characterizing Equation for the Derivatives