Pathwise Differentiability for SDEs in a Smooth Domain with Reﬂection

In this paper we study a Skorohod SDE in a smooth domain with normal reﬂection at the boundary, in particular we prove that the solution is pathwise differentiable with respect to the deterministic starting point. The resulting derivatives evolve according to an ordinary differential equation, when the process is in the interior of the domain, and they are projected to the tangent space, when the process hits the boundary.


Introduction
This paper contains a pathwise differentiability result for the solution (X t (x)) t≥0 of a stochastic differential equation (SDE) of the Skorohod type in a smooth bounded domain G ⊂ R d , d ≥ 2, with normal reflection at the boundary. The process (X t (x)) is driven by a d-dimensional standard Brownian motion and by a drift term, whose coefficients are supposed to be continuously differentiable and Lipschitz continuous, i.e. existence and uniqueness of the solution are ensured by the results of Lions and Sznitman in [18].
We prove that for every t > 0 the solution X t (x) is differentiable w.r.t. the deterministic initial value x in every direction v ∈ R d and we give a representation of the derivatives in terms of an ordinary differential equation. As an easy side result, we provide a Bismut-Elworthy formula for the gradient of the transition semigroup.
The resulting derivatives evolve according to a simple linear ordinary differential equation, when the process is away from the boundary, and they have a discontinuity and are projected to the tangent space, when the process hits the boundary. This evolution becomes rather complicated because of the structure of the set of times, when the process is at the boundary, which is known to be a.s. a closed set with zero Lebesgue measure without isolated points. However, this evolution does not give a complete characterization of the derivative process. Therefore, we establish a system of SDE-like equations, whose pathwise unique solution is the derivative process in coordinates w.r.t. a moving frame. This system is similar to the one introduced by Airault in [1] in order to develop probabilistic representations for the solutions of linear PDE systems with mixed Dirichlet-Neumann conditions in a smooth domain in R d . A further similar system appears in Section V.6 in [13], which deals with the heat equation for diffusion processes on manifolds with boundary conditions. This situation has also been considered in a more recent work by Hsu [12], where similar to [13] the associated matrixvalued Feynman-Kac multiplicative functional is constructed which is determined by the curvature tensor. The multiplicative functional associated with the pathwise derivatives obtained in this paper is very similar and possibly identical to the multiplicative functional in [12]. Nevertheless, the papers [1,12,13] deal with PDE systems or the heat equation, respectively, on smooth manifolds with mixed Neumann-Dirichlet boundary conditions, such that the solutions can be interpreted as the derivatives for the transition semigroup of the reflected Brownian motion. In this sense, our result can be considered as a pathwise version of the results in [1,12,13] with additional drift term.
In [11] Deuschel and Zambotti proved a pathwise differentiability result w.r.t. the initial data for diffusion processes in the domain G = [0, ∞) d . These results have already been transferred to SDEs in a convex polyhedron with possibly oblique reflection (see [3]). The proof of the main result in [11] is based on the fact that a Brownian path, which is perturbed by adding a Lipschitz path with a sufficiently small Lipschitz constant, attains its minimum at the same time as the original path (see Lemma 1 in [11]). This is due to the fact that a Brownian path leaves its minimum faster than linearly. In [11] this is used in order to provide an exact computation of the reflection term in the difference quotient via Skorohod's lemma.
Our approach is quite similar: Using localization techniques introduced by Anderson and Orey (cf. [2]) we transform the SDE locally into an SDE on a halfspace (cf. Section 2.3 below). Then, in order to compute the local time we need to deal with the pathwise minimum of a continuous martingale in place of the standard Brownian motion. Since the perturbations are now no longer Lipschitz continuous, i.e. Lemma 1 in [11] does not apply, and because of the asymptotics of a Brownian path around its minimum (cf. Lemma 3.7 below) one cannot necessarily expect that an analogous statement to Lemma 1 in [11] holds true in this case. Nevertheless, one can show that the minimum times converge sufficiently fast to obtain differentiability (see Proposition 3.8).
Another crucial ingredient in the proof is the Lipschitz continuity of the solution w.r.t. the initial data. This was proven by Burdzy, Chen and Jones in Lemma 3.8 in [6] for the reflected Brownian motion without drift in planar domains, but the arguments can easily be transferred into our setting (see Proposition 3.2). This will give pathwise convergence of the difference quotients along a subsequence. In order to identify the limit, we shall characterize the limit as the unique solution of the aforementioned SDE-like equation (cf. Section 4 in [1]).
A pathwise differentiability result w.r.t. the initial position of a reflected Brownian motion in smooth domains has also been proven by Burdzy in [4] using excursion theory. The resulting derivative is characterized as a linear map represented by a multiplicative functional for reflected Brownian motion, which has been introduced in Theorem 3.2 of [7]. In constrast to our main results, the SDE considered in [4] does not contain a drift term and the differentiability is shown for the trace process, while we consider the process on the original time-scale. However, we can recover the term, which is mainly characterizing the derivative in [4], describing the influence of curvature of ∂ G (cf. Remark 2.7 below).
In a series of papers [20,21,22] Pilipenko studies flow properties for SDEs with reflection and obtains Sobolev differentiability in the initial value, see [19] for a review of these results. In general, pathwise differentiability of diffusions processes w.r.t. the initial condition is a classical topic in stochastic analysis, see e.g. Theorem 4.6.5 in [17] for the case without reflection. On the other hand, reflected Brownian motions have been investigated in several articles, where the question of coalescence or noncoalescence of the two-point motion of a Brownian flow is of particular interest. For planar convex domains this has been studied by Cranston and Le Jan in [8] and [9], for some classes of non-smooth domains by Burdzy and Chen in [5], and for two-dimensional smooth domains by Burdzy, Chen and Jones in [6]. In higher dimension the case, where the domain is a sphere, has been considered by Sheu in [24] while the case of a general multi-dimensional smooth domain is still an open problem.
The paper is organized as follows: In Section 2 we give the precise setup and some further preliminaries and we present the main results. Section 3 is devoted to the proof of the main results.

General Notation
Throughout the paper we denote by . the Euclidian norm, by 〈., .〉 the canonical scalar product and by e = (e 1 , . . . , e d ) the standard basis in R d , d ≥ 2. Let G ⊂ R d be a connected closed bounded domain with C 3 -smooth boundary and G 0 its interior and let n(x), x ∈ ∂ G, denote the inner normal field. For any denote the orthogonal projection onto the tangent space. The closed ball in R d with center x and radius r will be denoted by B r (x). The transposition of a vector v ∈ R d and of a matrix A ∈ R d×d will be denoted by v * and A * , respectively. The set of continuous real-valued functions on G is denoted by C(G). For each k ∈ N, C k (G) denotes the set of real-valued functions that are k-times continuously differentiable in G. Furthermore, for f ∈ C 1 (G) we denote by ∇ the gradient of f and in the case where f is R d -valued by D f the Jacobi matrix. Finally, ∆ denotes the Laplace differential operator on C 2 (G) and D v := 〈v, ∇〉 the directional derivative operator associated with the direction v ∈ R d . The symbols c and c i , i ∈ N, will denote constants, whose value may only depend on some quantities specified in a particular context.

Skorohod SDE
For any starting point x ∈ G, we consider the following stochastic differential equation of the Skorohod type: where w is a d-dimensional Brownian motion on a complete probability space (Ω, , P) and l(x) denotes the local time of X (x) in ∂ G, i.e. it starts at zero, it is non-decreasing and it increases only at those times, when X (x) is at the boundary of G. The components b i : G → R of b are supposed to be in C 1 (G), in particular b is Lipschitz continuous. Then, existence and uniqueness of strong solutions of (2.1) are guaranteed by the results in [25] in the case, where G is a convex set, and for arbitrary smooth G by the results in [18]. The local time l(x) is carried by the set We define with the convention sup := 0. Then C is known to be a.s. a closed set of zero Lebesgue measure without isolated points. Note that t → r(t) is constant on each excursion interval of X (x) and is right-continuous. Moreover, for each t > inf C we have X r(t) (x) ∈ ∂ G.

Localization
In order to prove our main results we shall use the localization technique introduced in [ ii) There is a positive constant d 0 such that for every x ∈ G there exists an index m( iv) For every m ≥ 0 and i ∈ {1, . . . , d}, the functions and there exists a constant c, not depending on m and i, such that Note that these conditions imply n( Since ∂ G is supposed to be C 3 , the functions b i m and σ i m are continuously differentiable. We extend the functions b i m and σ i m to the whole domain G such that they are uniformly bounded and uniformly Lipschitz continuous on G. We will now define a sequence of stopping times (τ ) in such a way that on each interval [τ , τ +1 ) the process X (x) and small perturbations of it are confined to one of the coordinate patches U m . Fix now an arbitrary T > 0 and any δ 0 ∈ (0, d 0 ) and set∂ U m : Then, we define the sequence of stopping times (τ ) by where, for every ≥ 0, m := m(X τ (x)) ∈ N such that B d 0 (X τ (x)) ⊆ U m . Then τ ∈ C for every a.s. The dependence of τ on x will be suppressed in the notation. Note that by construction X t (x) ∈ U m for all t ∈ [τ , τ +1 ) and m = m +1 for every since δ 0 < d 0 . For abbreviation we set Using Itô's formula we get for t ∈ [τ , τ +1 ): For every we define a continuous semimartingale (M By the Girsanov Theorem there exists a probability measureP (x), which is equivalent to P and under which M x, is a continuous martingale. The quadratic variation process is given by which is strictly increasing in t on [τ , τ +1 ). We set ρ t := inf{s : [M x, ] s > t}. We can apply the Dambis-Dubins-Schwarz Theorem, in particular its extension in Theorem V.1.7 in [23], since in our Again we extend the functions O m to the whole domain G such that they are uniformly Lipschitz continuous on G.
Now we define the moving frame as the right-continuous process , which only jumps at the step times τ .
We apply Itô's formula locally on each interval [τ , τ +1 ) to obtain for some coefficient functions α k , β and γ depending on .

Main Results
Theorem 2.1. For all t > 0 and x ∈ G a.s. the mapping y → X t ( y) is directional differentiable at x, i.e. the limit (2.7) Remark 2.2. If x ∈ ∂ G, t = 0 is a.s. an accumulation point of C and we have r(t) > 0 a.s. for every t > 0. Therefore, in that case η 0 = v and η 0+ = π x (v), i.e. there is discontinuity at t = 0.

Remark 2.3.
The equation (2.7) does not characterize the derivatives, since it does not admit a unique solution. Indeed, if the process (η t ) solves (2.7), then the process (1 + l t (x))η t , t ≥ 0, also does. A characterizing equation for the derivatives is given Theorem 2.5 below.
Note that this result corresponds to that for the domain G = [0, ∞) d in Theorem 1 in [11]. The proof of Theorem 2.1 as well as the proofs of Theorem 2.5 and Corollary 2.9 below are postponed to Section 3. As soon as pathwise differentiability is established, we can immediately provide a Bismut-Elworthy formula: Define for all Corollary 2.4. For all f ∈ C(G), t > 0, x ∈ G and v ∈ R d we have: Proof. Formula (2.9) is straightforward from the differentiability statement in Theorem 2.1 and the chain rule. For formula (2.8) see the proof of Theorem 2 in [11].
From the representation of the derivatives in (2.7) it is obvious that (η t ) t evolves according to a linear differential equation, when the process X is in the interior of G, and that it is projected to the tangent space, when X hits the boundary. Furthermore, if X is at the boundary at some time t 0 and we have r(t 0 −) = r(t 0 ), i.e. t 0 is the endpoint of an excursion interval, then also η may have a discontinuity at t 0 and jump as follows: Consequently, we observe that at each time t 0 as above, η is projected to the tangent space and jumps in direction of n(X t 0 (x)) or −n(X t 0 (x)), respectively. Finally, if X t 0 (x) ∈ ∂ G and t → r(t) is continuous in t = t 0 , there is also a projection of η, but since in this case η t 0 − is in the tangent space, the projection has no effect and η is continuous at time t 0 .
where O t denotes the moving frame introduced in Section 2.3. Let P = diag(e 1 ) and Q = Id −P and Furthermore, we set γ 2 (t) := γ(X t (x)) · Q.

Theorem 2.5. There exists a right-continuous modification of η and Y , respectively, such that Y is characterized as the unique solution of
Remark 2.6. Here and in the sequel integrals of the form t r(t) ξ(s) d w s are understood as follows: The equations in Theorem 2.5 show that on every interval [τ , τ +1 ) the decomposition of Y is a decomposition into a discontinuous and continuous part. The discontinuous part Y 1 becomes zero whenever X hits the boundary, which corresponds to the projection of η to the tangent space described above. On the other hand, Y 2 is continuous which shows that only the component of η in normal direction is affected by the projection.
where for every x ∈ ∂ G, S(x) denotes the symmetric linear endomorphism acting on the tangent space at x, which is known as the shape operator or the Weingarten map, characterized by the relation S(x)v = −D v n(x) for all v in the tangent space at x. The eigenvalues of S(x) are the principal curvatures of ∂ G at x, and in two dimensions its determinant is the Gaussian curvature. Hence, the linear term in the equation for the derivatives in [4] can be recovered in our results. However, because of the presence of stochastic integrals in the characterizing equation in Theorem 2.5 it is unlikely that the result in [4] can be directly deduced from this equation.
Then, is a multiplicative functional that can possibly be identified with the discontinuous multiplicative functional constructed in [12] (cf. also [1,13]). Indeed, both functionals satisfy the same Bismut formula, see (2.9) and page 363 in [12]. Nevertheless, the evolution equation for the functional in Theorem 3.4 in [12] is slightly different from the one in Theorem 2.5, since in [12] the geometry of the domain is described in terms of horizontal lifts rather than in terms of a moving frame as in the present paper, which makes a direct identification difficult.
Finally, we give another confirmation of the results, namely they will imply that the Neumann condition holds for X .

Example: Processes in the Unit Disc
We end this section by considering the example of the unit disc to illustrate our results. Let the domain G = B 1 (0) be the closed unit disc in R 2 . Then, for x ∈ ∂ G, the inner normal field is given by n(x) = −x and the orthogonal projection onto the tangent space by π x (z) = z − 〈z, x〉x, z ∈ R 2 . The Skorohod equation can be written as and the system describing the derivatives becomes In this example a possible choice of the coordinate patches is the following. Let U 0 := G 0 be the interior of the disc and u 0 be the identity on U 0 . Further, for some small fixed δ we set and Finally, in order to define the moving frame, let O 0 := Id and from which the coefficient functions α 1 , α 2 and β can be defined accordingly. Note that in any case For simplicity we restrict ourselves now to the case b = 0. Then, the system in Theorem 2.5 can be rewritten as follows: we get in the case m = 0 that and in the case m = 0 that with initial value Y τ as specified in Theorem 2.5.

Lipschitz Continuity w.r.t. the Initial Datum
Before adressing the question of differentiability we establish pathwise continuity properties of x → (X t (x)) t w.r.t. the sup-norm topology. For this we will need that the mapping y → l t ( y) is bounded.
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 for all x, y ∈ G.
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 established in Lemma 3.1 Proof. The case x = y is clear and it suffices to consider the case T < inf{t : 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, Let T 0 := 0 and for k ≥ 1, Then, by Itô's formula we obtain for any k ≥ 1 and t ∈ (T k−1 , T k ], where we have used (3.1) and the Lipschitz continuity of b. Hence, for any t ∈ (T k−1 , T k ], which proves the proposition.

Remark 3.3. By Proposition 3.2 there exists for every
with δ 0 as in Section 2.3. Then, by the definition of τ we have for such y and for every that Proof. Fix T > 0 and x ∈ G and set 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 λ(x) as y tends to x.
For such y and s we get Hence, Using Proposition 3.2 and the fact that the functions σ m are uniformly Lipschitz the last term converges to zero as y tends to x. Recall that λ( y) converges uniformly on [τ , t] to λ(x) as y tends to x. Hence, we have that the associated signed measures dλ( y) on [τ , t] converge weakly to dλ(x) as y tends to x. Since s → σ 1 m (X s (x)) is bounded and continuous on [τ , t], the second term converges to zero as y tends to x. We apply the same argument for l τ ( y) − l τ (x) on [τ −1 , τ ] 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 The index x is there to indicate that the stopping times τ are the same as in the definition of M x, that are depending on x and not on . In particular, M x, ( ) 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 In the next lemma we show that M x, t ( ) is pathwise jointly continuous in t and .
Proof. In a first step we use Kolmogorov's continuity theorem to show the existence of a modification of (M ,x ( )) t, 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 Moreover, the functions b 1 m and σ 1 m are uniformly bounded and again by using Burkholder's inequality we get that for every p > 1 for some constant c 2 = c 2 (p, T ). Next we will show that for every p > 1 there exists a constant For the rest of the proof the symbol c denotes a constant whose value may change from one occurence to the other one.
By the uniform Lipschitz continuity of b m and Proposition 3.2 the first term can be estimated by 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: 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, ( )) t, 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.
where the right hand side is continuous in by Proposition 3.2 and Lemma 3.4.

Convergence of Minimum Times
In this section we investigate the behaviour of the local time, when the starting point x of X (x) has been perturbed by a small . To that purpose we shall transform the process locally into a process on the halfspace as indicated in Section 2.3, which allows us to use Skorohod's Lemma to compute the local time in terms of the time when the continuous martingale M x, attains its minimum. As a result we shall obtain that for tending to zero the minimum time of M x, converges almost surely faster than polynomially to the minimum time of M x, .
We fix from now on an arbitrary T > 0. In the following let (A n ) be the family of connected components of [0, T ]\C. Then, A n is open and recall that for every the mapping t → [M x, ] t is continuous and increasing on [τ , τ +1 ). Thus, for every n we may choose a random q n ∈ A n a follows: Let be such that inf In order to compute the local time l(x), recall that on every interval [τ , τ +1 ), ≥ 0, l(x) is carried by the set of times t, when u 1 m (X t (x)) = 0. Therefore, we can apply Skorohod's Lemma (see e.g. Lemma VI.2.1 in [23]) to equation (2.4) to obtain Fix any q n > inf C and such that q n ∈ [τ , τ +1 ) . Since u 1 m (X r(q n ) (x)) = 0 and t → L t (x) is non-decreasing, we have for all τ ≤ s ≤ r(q n ): where | | is always supposed to be sufficiently small, such that x lies in G. Furthermore, there exists a random ∆ n > 0 such that for all ∈ (−∆ n , ∆ n ) we have X t (x ) ∈ U m for all t ∈ [τ , q n ] (cf. Remark 3.3). As above we obtain for such : where r (q n ) is defined similarly as r(q n ). In particular, M x, Lemma 3.6. For all n we have r (q n ) → r(q n ) a.s. for → 0.
Proof. Consider some q n and let be such that q n ∈ [τ , τ +1 ). We fix now a typical ω such that r(q n ) is the unique time in [τ , q n ], when M x, attains its minimum and such that Lemma 3.5 holds. For every sequence ( k ) k converging to zero we can extract a subsequence of (r k (q n )), still denoted by (r k (q n )), converging to somer(q n ). By construction we have M x, for every k. Note that on one hand the right hand side converges to M x, r(q n ) as k → ∞ by Lemma 3.5. On the other hand the left hand side converges to M x, where the first term tends to zero for k → ∞ by Lemma 3.5 and the second term by the continuity of M x, . Thus, M x, r(q n ) . Since r(q n ) is unique time in [τ , q n ], when M x, attains its minimum, this impliesr(q n ) = r(q n ). Proof. It suffices to consider the case T = 1. We recall the following path decomposition of a Brownian motion, proven in [10]. Denoting by (M ,M ) two independent copies of the standard Brownian meander (see [23]), we set for all r ∈ (0, 1), Let now (τ, M ,M ) be an independent triple, such that τ has the arcsine law. Then, V τ d = W . This formula has the following meaning: τ is the unique time in [0, 1], when the path attains minimum − τM (1). The path starts in zero at time t = 0 and runs backward the path of M on [0, τ] and then it runs the path ofM . Moreover, it was proved in [14] that the law of the Brownian meander is absolutely continuous w.r.t. the law of the three-dimensional Bessel process (R t ) t≥0 on the time interval [0, 1] starting in zero. We recall that a.s.
for every function h satisfying the conditions in the statement (see [15], p. 164). Since the same asymptotics hold for the Brownian meander at zero, the claim follows.
In the next proposition we will apply Lemma 3.7 to the Brownian motions B x, defined in Section 2.3. More precisely, Lemma 3.7 gives that a.s. for every and every nonnegative q ∈ Q the following holds: If q ≤ [M x, ] τ +1 , denoting by ϑ q the unique time when B x, attains its minimum over [0, q], B x, satisfies the asymptotic behaviour stated in (3.5) at ϑ q .

Proposition 3.8. Let δ > 0 be arbitrary. Then, for all n we have
Proof. First we fix 0 < δ < 1. By construction we have for every q n and such that q n ∈ [τ , τ +1 ) and for small enough, M x, Since M x, t ( , 0) as in Lemma 3.5, this can be rewritten as M x, , where B x, is aP (x)-Brownian motion (see (2.5)) and B x, attains its . Note that q n has been chosen such that [M x, ] q n ∈ Q. Hence, applying Lemma 3.7 with h(t) = t δ /2 it follows that a.s.
for all ∈ (−∆ n , ∆ n ) for some positive ∆ n . Since we have by Lemma 3.6, possibly after choosing a smaller ∆ n , that σ 1 m (X r (x)) 2 is bounded away from zero uniformly in r between r(q n ) and r (q n ). Thus, and we derive from (3.6) that a.s.

Corollary 3.9.
For any n and let be such that q n ∈ [τ , τ +1 ). Then, Proof. For arbitrary δ ∈ (0, 1 2 ) we have is a.s bounded by a random constant due to Lemma 3.5 and due to the fact that M x, is Hölder continuous of order 1 2 − δ, we obtain i) from Proposition 3.8. ii) follows from i). Indeed, by Proposition 3.2 and Lemma 3.6 we have for sufficiently small that l r(q n ) (x ) = l r (q n ) (x ) = 0 if q n < inf C and l r(q n ) (x ), l r (q n ) (x ) > 0 if q n > inf C. In the first case ii) is trivial and the latter case we have by (3.4) where we have used the fact that M x, ( ) attains its minimum over [τ , q n ] at time r (q n ) and its minimum over [τ , r(q n )] at time r (r(q n )), respectively.

Proof of the Differentiability
The main idea in order to prove the differentiability result is based on a pathwise argument similar to Step 5 in the proof of Theorem 1 in [11]. Denote by η t ( ) := 1 (X t (x ) − X t (x)) the difference quotient, x = x + v for any fixed vector v ∈ R d and let T > 0 be fixed. Then, the first step is to prove the following Proposition 3.10. There exists a set Ω 0 ⊆ Ω with P[Ω 0 ] = 1 such that for every ω ∈ Ω 0 the following holds. Let ( ν ) ν = ( ν (ω)) ν be any random sequence such that lim ν→∞ ν (ω) = 0 for every ω ∈ Ω 0 . Then, there exists a subsequence ( ν l ) l with (ν l ) l = (ν l (ω)) l such that for all t ∈ [0, T ]\C(ω) the limit of η t ( ) along the subsequence ( ν l ) l , i.e. lim l→∞ η t (ω, ν l (ω) (ω)) =:η t (ω, ( n )) =:η t (ω), exists and is measurable. Furthermore, for all ω ∈ Ω 0 , (η t ) t∈[0,T ]\C satisfieŝ (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 0 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 extendη(ω), ω ∈ Ω 0 , to a right-continuous trajectory on [0, T ]. Then, we prove that in coordinates of the moving frame introduced in Section 2.2η 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 ( ) converges a.s. if and only if for every component Let now ( i,− ν ) ν and ( i,+ ν ) ν be two random sequences, along which η i T ( ) converges to its limes inferior and its limes superior, respectively. We apply Proposition 3.10 to these two sequences and get two limiting objectsη − andη + , being two trajectories in R d whose i-th components at time T ,η

The Limit along a Subsequence
Proof of Proposition 3.10. Let T > 0 still be fixed. We choose Ω 0 ⊆ Ω with full measure such that for all ω ∈ Ω 0 Lemma 3.1 holds and Corollary 3.9 holds for all n. Let now ω ∈ Ω 0 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 ) = 0 if q n < inf C and both of them are strictly positive if q n > inf C for all | | ∈ (0, ∆ n ). Then, where X α, r := αX r (x ) + (1 − α)X r (x), α ∈ [0, 1], and n(X r (x )) d l r (x ). (3.11) Note that if q n < inf C we have r(q n ) = 0, η r(q n ) ( ) = v and R q n (x ) = 0. In any case, by Corollary 3.9. Recall that η t ( ) ≤ exp(c 1 (T + l T (x) + l T (x ))) ≤ c for all t ∈ [0, T ] and = 0 by Proposition 3.2 and Lemma 3.1. Let ( ν ) ν = ( ν (ω)) ν be any random sequence converging to zero. By a diagonal procedure, we can extract a subsequence ( ν l ) l with (ν l ) l = (ν l (ω)) l such that η r(q n ) ( ν l ) has a limitη r(q n ) ∈ R d as l → ∞ for all n ∈ N.
Let nowη : [0, T ]\C → R d be the unique solution of By (3.10) and Proposition 3.2, we get for | | ∈ (0, ∆ n ) and t ∈ A n , Since η r(q n ) ( ν l ) →η r(q n ) , R q n (x ) → 0 , X α, ν l 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 ( ν l ) converges toη t uniformly in t ∈ A n for every n. Thus, since C has zero Lebesgue measure, η t ( ν l ) converges toη t for all t ∈ [0, T ]\C as l → ∞ and by the dominated convergence theorem we get that (η t (ω)) t∈[0,T ]\C satisfies (3.8). Finally, the measurability ofη is immediate from its construction.
From now on we will denote byη the limiting object constructed in Proposition 3.10 from any arbitrary but fixed random sequence ( n ). The next lemma shows thatη r(q n ) is in the tangent at X r(q n ) (x) for every q n . Lemma 3.11. For every q n > inf C, i) 〈η r(q n ) ( ), n(X r(q n ) (x))〉 → 0 a.s. and in L p , p > 1, as → 0, ii) 〈η r (q n ) ( ), n(X r (q n ) (x ))〉 → 0 a.s. and in L p , p > 1, as → 0.
Note that the term of second order in the Taylor expansion is in O( ) by Proposition 3.2. Recall that u 1 m (X r(q n ) (x)) = 0, and combining formula (2.4) and (3.4), we get for all ∈ (−∆ n , ∆ n ) for some positive ∆ n . Arguing similarly as in (3.7) we obtain from Corollary 3.9 i) that and i) follows. The proof of ii) is rather analogous. For an appropriate ∆ n > 0 we have r (q n ) ∈ [τ , τ +1 ) and l r (q n ) (x) > 0 for all | | ∈ (0, ∆ n ). Then, for such we get again by using Taylor's formula and the fact that u 1 m (X r (q n ) (x )) = 0, Since M x, attains its minimum over [τ , q n ] at time r(q n ) and its minimum over [τ , r (q n )] at time r(r (q n )), respectively, we finally get |〈η r (q n ) ( ), n(X r (q n ) (x ))〉| ≤ 1 which tends to zero again by Corollary 3.9 i).
So farη t is not defined for every t ∈ C, only for the left endpoints r(q n ) of the excursion intervals.
We will now extend the trajectories ofη 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 For that purpose it is sufficient to define 〈η t , ∇u i m (X t (x))〉 for i ∈ {1, . . . , d}. We set 〈η t , ∇u 1 m (X t (x))〉 =〈η t , n(X t (x))〉 := 0 and 〈η t , ∇u i m (X t (x))〉 := I * i (t) for i = 2, . . . , d. Having extendedη to a trajectory on [0, T ] we can defineÎ i (t), t ∈ [0, T ], similarly to I * i (t) locally on each interval [τ , τ +1 ). Note that I * i andÎ i are continuous on each interval [τ , τ +1 ), in particular they are right-continuous on [0, T ], and since C has zero Lebesgue measure, for every j = 1, . . . , d, where as before X α, Proof. By Proposition 3.2 and Lemma 3.1 the first term can be estimated as follows For the second term we get similarly, using Burkholder's inequality, Hence both terms converges to zero along ( ν k ) by dominated convergence, since X ii) For every i ∈ {2, . . . , d} we have for all t ∈ [0, T ] with such that t ∈ [τ , τ +1 ).
In particular, the trajectories ofη are right-continuous on [0, T ].
Proof. i) First note that sinceÎ 1 is continuous on each interval [τ , τ +1 ) and t → r(t) is rightcontinuous, the paths of are also right-continuous on [0, T ]. Let t ∈ [0, T ] be fixed, be such that t ∈ [τ , τ +1 ) and ∆ T > 0 be as in Remark 3.3. Then, t ∈ C a.s. Further, if t < inf C and | | < ∆ T then l t (x) = l τ (x) and l t (x ) = l τ (x ) . So we have by Taylor's formula and (2.2) that where as before X α, where the first term and the last term converge to zero by Corollary 3.9ii) and Lemma 3.11, respectively. The remaining terms converge in L 2 to the corresponding terms in the definition ofÎ 1 by Lemma 3.12. Hence, we obtain that F 1 (t) = (t) a.s. for every t. Since the trajectories ofη 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 . Hence, Finally, since by definition F 1 = 〈η . , ∇u i m (X . (x))〉 = 0 = on C ∩ [τ , τ +1 ) for every , we obtain which gives i).
ii) We proceed similarly to i). Let i ∈ {2, . . . , d} and t ∈ [0, T ] be fixed and be such that t ∈ [τ , τ +1 ). Then, t ∈ C a.s. and we have by Taylor's formula and (2.2) 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) =Î i (t) = I * i (t) a.s. for every t. Since the trajectories ofη are continuous on every excursion interval A n , the path of F i are right-continuous on every A n and we get Finally, since by definition F i = 〈η . , ∇u i m (X . (x))〉 = I * i on C ∩ [τ , τ +1 ), we use (3.13) to obtain and we obtain ii).
The right-continuity of the trajectories ofη is now immediate from i) and ii). Indeed, writingη t in the basis n m (X t (x)), we get that on one hand the basis vectors are continuous in t on [τ , τ +1 ) and on the other hand the coordinates are right-continuous in t by i) and ii).
The extension ofη on C and Lemma 3.11 imply that 〈η t , ∇u 1 m (X t (x))〉 = 〈η t , n(X t (x))〉 = 0 for all t ∈ [τ , τ +1 ) ∩ C, i.e. when the process X (x) is at the boundaryη is at the tangent space, while the projection ofη is a continuous process as indicated by (3.14).
Let now for all x ∈ U m , m ≥ 0 and η ∈ R d For later use we prove now uniform convergence ofΠ m X t (x ) (η t ( )) toΠ m X t (x) (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 , i = 2, . . . , d. In particular, note thatΠ m X t (x) (η t ) is not the same as Q · O t ·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 , τ +1 ) ∩ C. Lemma 3.14. Let ∆ T > 0 be as in Remark 3.3 such that, for every ≥ 0,Π m X s (x ) (η s ( )) is well defined for all s ∈ [τ , τ +1 ) and all 0 < | | < ∆ T . Then, and for this it is enough to prove that for every i ∈ {2, . . . , d}, For | | < ∆ T we use as before Taylor's formula and (2.2) to obtain where again X α, Comparing (3.14) and (3.16) leads to The claim follows now from Lemma 3.12 and the fact that η τ ( ν k ) →η τ .

A Characterizing Equation for the Derivatives
The next step to prove the differentiability result is to identify the derivative. To that aim we shall establish a system of SDE-like equations, which admits a unique solution and which is solved bŷ Y t := O t ·η t , t ∈ [0, T ], O t denoting the moving frame defined in Section 2.3. In other words, we shall show thatŶ is the unique solution of the system in Theorem 2.5.
We shall proceed similarly to Section 4 in [1] (see also Section V.6 in [13]), namely we shall derive an equation for Y t ( ) := O t · η t ( ), t ∈ [0, T ], which converges in L 2 to the equation in Theorem 2.5. However, it is a general principle in the theory of stochastic differential equations that if pathwise uniqueness holds, then any 'reasonable' approximation converges at least in probability to the solution (see [16]).
Let the rows of O t be denoted by n k t = n k (X t (x)), k = 1, . . . d. Then, we obtain by the chain rule that for every t where again X 1]. By applying Itô's integration by parts formula on each interval [τ , τ +1 ) we get with coefficient functions α k and β and γ as in (2.6). Let P = diag(e 1 ) and Q = Id −P and set to decompose the space R d into the direct sum Im P ⊕ Ker P. We define the coefficients c and d to be such that As before let t ∈ [0, T ]\C and let n and be such that t ∈ A n ∩ [τ , τ +1 ). Then we choose ∆ T > 0 as in Remark 3.3 and such that a.s. l q n (x) = l q n (x ) = 0 if q n < inf C and both of them are strictly positive if q n > inf C for all 0 < | | < ∆ T . For such we get with R q n (x ) as in (3.11). Moreover, since η t ( ) is continuous in t, the initial value is given by Next we compute the corresponding equation for Y 2, . For s ∈ [τ , t] let the rows of O m (X s (x )) be denoted by n k (X s (x )), k = 1, . . . d. In particular, for k ∈ {2, . . . , d} we have n k (X s (x)) · n(X s (x)) d l s (x) = 0 and n k (X s (x )) · n(X s (x )) d l s (x ) = 0. For such k we use again Taylor's formula to obtain n k (X s (x)) · 1 n(X s (x )) d l s (x ) − n(X s (x)) d l s (x) = 1 n k (X s (x)) · n(X s (x )) d l s (x ) = 1 (n k (X s (x)) − n k (X s (x ))) · n(X s (x )) d l s (x ) = − η s ( ) * · Dn k (X s (x )) * · n(X s (x )) d l s (x ) + O( ) =η s ( ) * · (Dn(X s (x ))) * · n k (X s (x )) * d l s (x ) + O( ) =n k (X s (x )) · Dn(X s (x )) · η s ( ) d l s (x ) + O( ) =n k (X s (x )) · Dn(X s (x )) · O −1 s · Y s ( ) d l s (x ) + O( ). Hence, with γ 1 (s) := γ(X s (x)) · P and γ 2 (s) := γ(X s (x)) · Q. The initial value is given by for ≥ 1 and Y 2, , the next step is to prove the following Obviously, from the second equation in Theorem 2.5 it follows thatŶ 2 t is a continuous semimartingale in t on every interval [τ , τ +1 ). Hence, the mapping t → π X r(q n ) (x) (η t ) is continuous at time t = r(q n ) for every n. To complete the proof of Theorem 2.1 we need to show Proposition 3.15 and that the system in Theorem 2.5 admits a unique solution. First, we prove two preparing lemmas. Proof. Since Φ 1 is uniformly bounded, we get where the second term tends to zero by Proposition 3.2. Let now σ s := inf{r : l r (x ) ≥ s} be the left-continuous inverse of l(x ). For any fixed s > 0 we have a.s. X s (x) ∈ ∂ G and by Proposition 3.2 X s (x ) ∈ ∂ G if | | < ∆ s for some positive random ∆ s . Hence, for such , σ s is a.s. the left endpoint of an excursion interval of X (x ). In particular, σ s = r (q n ) a.s. for some q n depending on s. Then, by Lemma 3.11ii) we get E 〈η σ s ( ), n(X σ s (x ))〉 2 1l {| |<∆ s } = E 〈η r (q n ) ( ), n(X r (q n ) (x ))〉 2 1l {| |<∆ s } → 0, as → 0. Thus, E 〈η σ s ( ), n(X σ s (x ))〉 2 tends to zero for every s, so we can apply the domi- We show now that also the first term in (3.20) tends to zero. On one hand, we use the change of variables formula for Stieltjes integrals (see e.g. Proposition 4.9 in Chapter 0 in [23]) to obtain for an arbitrary M > 0 Finally, we let M tend to infinity and obtain that i) holds. ii) follows by an analogous, simpler proceeding. Note that the finiteness of the exponential moments of the local time, which is needed in the final step, can be deduced for instance from (2.4). Indeed, the stochastic integral does have finite exponential moments and the remaining terms are uniformly bounded.
So far Φ(t) and Φ (t) are only defined on the support of l(x) and l(x ), respectively. For the next proof we extend them to the whole interval [0, T ] by setting if t ∈ [τ , τ +1 ) and we define Φ 1 (t), Φ 2 (t) as well as Φ 1 (t) and Φ 2 (t) as before.