Lipschitz Continuity w.r.t. the Initial Datum

Finally, in order to define the moving frame, let O := Id and O m x := 1 kxk ‚ −x 1 −x 2 −x 2 x 1 Œ := O −1 m x, m = 1, . . . , 4. Hence, dO t · O −1 t = 0 on [τ ℓ , τ ℓ+1 if m ℓ = 0, and otherwise we use Itô’s formula to obtain dO t ·O −1 t = X 2 t x kX t xk 3 ‚ 1 −1 0 Œ d w 1 t + X 1 t x kX t xk 3 ‚ −1 1 Œ d w 2 t + – X 2 t x kX t xk 3 ‚ 1 −1 0 Œ b 1 X t x + X 1 t x kX t xk 3 ‚ −1 1 Œ b 2 X t x + 1 2 kX t xk 4 ‚ 1 0 0 1 Œ™ d t, from which the coefficient functions α 1 , α 2 and β can be defined accordingly. Note that in any case γ = 0. Furthermore, since nx = −x for all x ∈ ∂ G, Dnx = − Id and therefore Φ 2 t = −Q. For simplicity we restrict ourselves now to the case b = 0. Then, the system in Theorem 2.5 can be rewritten as follows: For t ∈ [τ ℓ , τ ℓ+1 , writing Y t = y 1 t , y 2 t , we get in the case m ℓ = 0 that y 1 t = 1l {tinf C ℓ } y 1 τ ℓ , y 2 t = y 2 τ ℓ − Z t τ ℓ y 2 s d l s x, and in the case m ℓ 6= 0 that y 1 t =1l {tinf C ℓ } y 1 τ ℓ + Z t τ ℓ X 2 s x kX s xk 3 y 2 s d w 1 s − Z t τ ℓ X 1 s x kX s xk 3 y 2 s d w 2 s + Z t τ ℓ 1 2 kX s xk 4 y 1 s ds + 1l {t≥inf C ℓ } Z t rt X 2 s x kX s xk 3 y 2 s d w 1 s − Z t rt X 1 s x kX s xk 3 y 2 s d w 2 s + Z t rt 1 2 kX s xk 3 y 1 s ds y 2 t = y 2 τ ℓ − Z t τ ℓ X 2 s x kX s xk 3 y 1 s d w 1 s + Z t τ ℓ X 1 s x kX s xk 3 y 1 s d w 2 s + Z t τ ℓ 1 2 kX s xk 4 y 2 s ds − Z t τ ℓ y 2 s d l s x, with initial value Y τ ℓ as specified in Theorem 2.5. 3 Proof of the Main Result

3.1 Lipschitz Continuity w.r.t. the Initial Datum

Before adressing the question of differentiability we establish pathwise continuity properties of x 7→ X t x t w.r.t. the sup-norm topology. For this we will need that the mapping y 7→ l t y is bounded. Lemma 3.1. For every t 0 we have sup x ∈G l t x ∞ a.s. Proof. See Lemma 3.3 in [4]. Proposition 3.2. Let T 0 be arbitrary and let X t x and X t y, t ≥ 0, be two solutions of 2.1 for any x, y ∈ G. Then, there exists a positive constant c only depending on T such that sup t ∈[0,T ] kX t x − X t yk ≤ kx − yk expcT + l T x + l T y for all x, y ∈ G. 854 Note that the Lipschitz continuity in the initial condition, which is stated here, becomes effective since the Lipschitz constant can be controlled due to the uniform boundedness of l T x in x estab- lished in Lemma 3.1 Proof. The case x = y is clear and it suffices to consider the case T inf{t : X t x = X t y}. We shall proceed similarly to Lemma 3.8 in [6]. Since ∂ G is C 2 -smooth and G is connected and compact, there exists a positive constant c 1 ∞ such that for all x ∈ ∂ G and all y ∈ G, 〈x − y, nx〉 ≤ c 1 kx − yk 2 . 3.1 Let T := 0 and for k ≥ 1, T k := inf ¦ t ≥ T k −1 : kX t x − X t yk 6∈ € 1 2 kX T k −1 x − X T k −1 yk, 2kX T k −1 x − X T k −1 yk Š© ∧ T. Then, by Itô’s formula we obtain for any k ≥ 1 and t ∈ T k −1 , T k ], kX t x − X t yk − kX T k −1 x − X T k −1 yk = Z t T k −1 X r x − X r y, bX r x − bX r y kX r x − X r yk d r + Z t T k −1 X r x − X r y, nX r x kX r x − X r yk d l r x + Z t T k −1 X r y − X r x, nX r y kX r x − X r yk d l r y ≤c 2 Z t T k −1 kX r x − X r yk d r + c 1 Z t T k −1 kX r x − X r yk dl r x + d l r y ≤c 3 kX T k −1 x − X T k −1 yk Z T k T k −1 d r + d l r x + d l r y, where we have used 3.1 and the Lipschitz continuity of b. Hence, for any t ∈ T k −1 , T k ], kX t x − X t yk kX T k −1 x − X T k −1 yk ≤ 1 + c 3 € T k − T k −1 + l T k x − l T k −1 x + l T k y − l T k −1 y Š ≤ exp € c 3 € T k − T k −1 + l T k x − l T k −1 x + l T k y − l T k −1 y ŠŠ , and kX t x − X t yk kx − yk = kX t x − X t yk kX T k −1 x − X T k −1 yk k −1 Y j=1 kX T j x − X T j yk kX T j −1 x − X T j −1 yk ≤ k Y j=1 exp c 3 T j − T j −1 + l T j x − l T j −1 x + l T j y − l T j −1 y ≤ exp € c 3 € T k + l T k x + l T k y ŠŠ ≤ exp c 3 T + l T x + l T y , which proves the proposition. 855 Remark 3.3. By Proposition 3.2 there exists for every T 0 a random ∆ T 0 such that sup t ∈[0,T ] kX t x − X t yk δ 2 , ∀ y ∈ B ∆ T x ∩ G, with δ as in Section 2.3. Then, by the definition of τ ℓ we have for such y and for every ℓ that X t y ∈ U m ℓ for all t ∈ [τ ℓ , τ ℓ+1 . Lemma 3.4. For every t ∈ [0, T ] we have that for all x ∈ G the mapping y 7→ l t y is continuous at x. Proof. Fix T 0 and x ∈ G and set λ t y := Z t nX r y d l r y, y ∈ G, t ∈ [0, T ], which defines for each y ∈ G a process of bounded variation on [0, T ]. Then, we get immediately by Proposition 3.2, 2.1 and the Lipschitz property of b that λ y converges uniformly on [0, T ] to λx as y tends to x. Let now t ∈ [0, T ] and ℓ be such that t ∈ [τ ℓ , τ ℓ+1 . Then, for all y ∈ B ∆ T x ∩ G with ∆ T as in Remark 3.3 we have that X s y, X s x ∈ U m ℓ for all s ∈ [τ ℓ , t. For such y and s we get d l s y − dl s x = 〈∇u 1 m ℓ X s y, nX s y〉 dl s y − 〈∇u 1 m ℓ X s x, nX s x〉 dl s x = ∇u 1 m ℓ X s y dλ s y − ∇u 1 m ℓ X s x dλ s x = σ 1 m ℓ X s y dλ s y − σ 1 m ℓ X s x dλ s x. Hence, l t y − l t x =l τ ℓ y − l τ ℓ x + Z t τ ℓ σ 1 m ℓ X s x dλ s y − dλ s x + Z t τ ℓ σ 1 m ℓ X s y − σ 1 m ℓ X s x d λ s y. Using Proposition 3.2 and the fact that the functions σ m are uniformly Lipschitz the last term con- verges to zero as y tends to x. Recall that λ y converges uniformly on [τ ℓ , t] to λx as y tends to x. Hence, we have that the associated signed measures d λ y on [τ ℓ , t] converge weakly to d λx as y tends to x. Since s 7→ σ 1 m ℓ X s x is bounded and continuous on [τ ℓ , t], the second term converges to zero as y tends to x. We apply the same argument for l τ ℓ y − l τ ℓ x on [τ ℓ−1 , τ ℓ ] and by iterating this procedure we obtain the claim. We fix now an arbitrary x ∈ G, v ∈ R d and T 0. Then, we set x ǫ := x + ǫ v for all ǫ ∈ [a x , b x ] with a x ≤ 0 and b x ≥ 0 such that x ǫ ∈ G for all ǫ ∈ [a x , b x ]. Furthermore, we define for such ǫ and t ∈ [0, T ], M x, ℓ t ǫ :=    if t ∈ [0, τ ℓ , R t τ ℓ b 1 m ℓ X r x ǫ d r + R t τ ℓ σ 1 m ℓ X r x ǫ d w r if t ∈ [τ ℓ , τ ℓ+1 ], M x, ℓ τ ℓ+1 ǫ if t τ ℓ+1 . 856 The index x is there to indicate that the stopping times τ ℓ are the same as in the definition of M x, ℓ that are depending on x and not on ǫ. In particular, M x, ℓ ǫ is a well-defined object, since the coefficient functions b 1 m and σ 1 m have been extended to the whole domain G. Note that M x, ℓ t = M x, ℓ t 0, t ∈ [0, T ]. Finally, we set ∆M x, ℓ t ǫ, ǫ ′ := M x, ℓ t ǫ − M x, ℓ t ǫ ′ , t ∈ [0, T ], ǫ, ǫ ′ ∈ [a x , b x ], so that ∆M x, ℓ t ǫ, 0 = Z t τ ℓ b 1 m ℓ X r x ǫ − b 1 m ℓ X r x d r + Z t τ ℓ σ 1 m ℓ X r x ǫ − σ 1 m ℓ X r x d w r , for t ∈ [τ ℓ , τ ℓ+1 and ǫ ∈ [a x , b x ]. In the next lemma we show that M x, ℓ t ǫ is pathwise jointly continuous in t and ǫ. Lemma 3.5. Let ∆ T be as in Remark 3.3. Then, for a.e. ω ∈ Ω the following holds. For every δ 1 ∈ 0, 1 and δ 2 ∈ 0, 1 2 there exists a random constant K = Kω, δ 1 , δ 2 , T such that ∆ M x, ℓ t ǫ, ǫ ′ − ∆M x, ℓ s ǫ, ǫ ′ ≤ K |ǫ − ǫ ′ | 1 −δ 1 |t − s| 1 2 −δ 2 , ∀s, t ∈ [0, T ], 3.2 for all ǫ, ǫ ′ such that x ǫ , x ǫ ′ ∈ B ∆ T x ∩ G. In particular, for every δ ∈ 0, 1 we have for all such ǫ M x, ℓ t ǫ − M x, ℓ t ≤ K |ǫ| 1 −δ , ∀t ∈ [0, T ], for some random constant K = K ω, δ, T . Proof. In a first step we use Kolmogorov’s continuity theorem to show the existence of a modification of M ℓ,x ǫ t, ǫ satisfying the above estimate and in a second step we show the claim by a continuity argument. Step 1: It follows directly from Proposition 3.2, the uniform Lipschitz continuity of b 1 m and σ 1 m and the Burkholder inequality that for every p 1 there exists a positive constant c 1 = c 1 p, T such that E • M x, ℓ t ǫ − M x, ℓ t ǫ ′ p ˜ ≤ c 1 |ǫ − ǫ ′ | p , ∀t ∈ [0, T ], ǫ, ǫ ′ ∈ [a x , b x ]. Moreover, the functions b 1 m and σ 1 m are uniformly bounded and again by using Burkholder’s inequal- ity we get that for every p 1 E • M x, ℓ t ǫ − M x, ℓ s ǫ p ˜ ≤ c 2 |t − s| p 2 , ∀s, t ∈ [0, T ], ǫ ∈ [a x , b x ] for some constant c 2 = c 2 p, T . Next we will show that for every p 1 there exists a constant c 3 = c 3 p, T such that E • ∆ M x, ℓ t ǫ, ǫ ′ − ∆M x, ℓ s ǫ, ǫ ′ p ˜ ≤ c 3 |ǫ − ǫ ′ | p |t − s| p 2 , ∀s, t ∈ [0, T ], ǫ, ǫ ′ ∈ [a x , b x ]. For the rest of the proof the symbol c denotes a constant whose value may change from one oc- curence to the other one. Let 0 ≤ s ≤ t ≤ T and ǫ, ǫ ′ ∈ [a x , b x ]. Recall that both M x, ℓ ǫ and 857 M x, ℓ ǫ ′ are defined to be constant on [0, T ]\[τ ℓ , τ ℓ+1 ]. Thus, it is enough to consider the case where [s, t] intersects [ τ ℓ , τ ℓ+1 ]. Setting ˆ s i := s ∨ τ ℓ and ˆt := t ∧ τ ℓ+1 we have |ˆt − ˆs| ≤ |t − s|. By the definition of M x, ℓ ǫ and M x, ℓ ǫ ′ we have E • ∆ M x, ℓ t ǫ, ǫ ′ − ∆M x, ℓ s ǫ, ǫ ′ p ˜ ≤c E   Z ˆt ˆ s b 1 m ℓ X r x ǫ − b 1 m ℓ X r x ǫ ′ d r p   + c E   Z ˆt ˆ s σ 1 m ℓ X r x ǫ − σ 1 m ℓ X r x ǫ ′ d w r p   . By the uniform Lipschitz continuity of b m and Proposition 3.2 the first term can be estimated by c |t − s| p E – sup r ∈[ˆs,ˆt] kX r x ǫ − X r x ǫ ′ k p ™ ≤ c |ǫ − ǫ ′ | p |t − s| p . For the second term we get the following estimate by Burkholder’s inequality, the uniform Lipschitz continuity of σ m and again by Proposition 3.2: c E   sup r ∈[ˆs,ˆt] Z r ˆ s σ 1 m ℓ X r x ǫ − σ 1 m ℓ X r x ǫ ′ d w r p   ≤c E    Z ˆt ˆ s kX r x ǫ − X r x ǫ ′ k 2 d r p 2    ≤c |t − s| p 2 E – sup r ∈[ˆs,ˆt] kX r x ǫ − X r x ǫ ′ k p ™ ≤c |ǫ − ǫ ′ | p |t − s| p 2 and we obtain the desired estimate. We apply now Kolmogorov’s continuity theorem, in particular the version for double parameter random fields in Theorem 1.4.4 in [17], which implies that for any given δ 1 ∈ 0, 1 and δ 2 ∈ 0, 1 2 there exists a modification of the random field M x, ℓ ǫ t, ǫ satisfying 3.2 for some random constant K = K ω, δ 1 , δ 2 , T . Step 2: The existence of a modification shown in Step 1 immediately implies that a.s. 3.2 holds for all s, t ∈ [0, T ] ∩ Q and ǫ, ǫ ′ ∈ [a x , b x ] ∩ Q. The claim follows if ∆M x, ℓ t ǫ, ǫ ′ − ∆M x, ℓ s ǫ, ǫ ′ is pathwise continuous in s and t as well as in ǫ and ǫ ′ . It is enough to show the continuity of M x, ℓ t ǫ in t and ǫ. The continuity in t is obvious and for every ǫ such that x ǫ ∈ B ∆ T x ∩ G we get by an application of Itô’s formula as in 2.2 M x, ℓ t ǫ = u 1 m ℓ X t x ǫ − u 1 m ℓ X τ ℓ x ǫ − L t x ǫ , where the right hand side is continuous in ǫ by Proposition 3.2 and Lemma 3.4.

3.2 Convergence of Minimum Times