getdoc9842. 329KB Jun 04 2011 12:04:43 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y}

Vol. 16 (2011), Paper no. 28, pages 845–879. Journal URL

http://www.math.washington.edu/~ejpecp/

Pathwise Differentiability for SDEs in a Smooth Domain

with Reflection

Sebastian Andres

∗

Abstract

In this paper we study a Skorohod SDE in a smooth domain with normal reflection at the bound-ary, in particular we prove that the solution is pathwise differentiable with respect to the de-terministic 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.

Key words:Stochastic differential equation with reflection, normal reflection, local time. AMS 2000 Subject Classification:Primary 60J55, 60H10.

Submitted to EJP on May 11, 2010, final version accepted March 27, 2011.

∗_{University of British Columbia, 121-1984 Mathematics Road Vancouver, B.C. V6T 1Z2, Canada;}

1

Introduction

This paper contains a pathwise differentiability result for the solution(X_t(x))_t≥0of a stochastic dif-ferential equation (SDE) of the Skorohod type in a smooth bounded domainG ⊂Rd, d ≥2, with normal reflection at the boundary. The process(X_t(x))is driven by ad-dimensional standard Brow-nian 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 everyt>0 the solutionX_t(x)is differentiable w.r.t. the deterministic initial value

x in every directionv∈Rd 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 inRd. 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 matrix-valued 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 domainG= [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.

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 sub-sequence. 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 gen-eral, 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 inter-est. 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 prelimi-naries and we present the main results. Section 3 is devoted to the proof of the main results.

2

Main Results and Preliminaries

2.1

General Notation

Throughout the paper we denote by _k._k the Euclidian norm, by_〈., ._〉 the canonical scalar product and bye= (e1, . . . ,ed)the standard basis inRd, d≥2. LetG⊂Rd be a connected closed bounded domain withC3-smooth boundary andG₀its interior and letn(x), x_∈∂G, denote the inner normal field. For anyx _∈∂G, let

πx(z):=z− 〈z,n(x)〉n(x), z∈Rd,

denote the orthogonal projection onto the tangent space. The closed ball inRd with center x and radius r will be denoted byB_r(x). The transposition of a vectorv _∈Rd and of a matrixA_∈Rd×d

denoted byC(G). For each k_∈N, Ck(G)denotes the set of real-valued functions that are k-times continuously differentiable inG. Furthermore, for f ∈C1(G)we denote by_∇the gradient of f and in the case wheref isRd-valued byD f the Jacobi matrix. Finally,∆denotes the Laplace differential operator onC2(G)andD_v:=_〈v,_∇〉the directional derivative operator associated with the direction

v∈Rd. The symbolscandci,i∈N, will denote constants, whose value may only depend on some

quantities specified in a particular context.

2.2

Skorohod SDE

For any starting point x _∈G, we consider the following stochastic differential equation of the Sko-rohod type:

Xt(x) =x+

Z t

0

b(Xr(x))d r+wt+

Z t

0

n(Xr(x))d lr(x), t≥0,

X_t(x)_∈G, d l_t(x)_≥0,

Z ∞

0

1l_G0(X_t(x))d l_t(x) =0, t≥0,

(2.1)

where w is a d-dimensional Brownian motion on a complete probability space (Ω,_F,P) andl(x)

denotes the local time ofX(x)in∂G, i.e. it starts at zero, it is non-decreasing and it increases only at those times, whenX(x)is at the boundary ofG. The componentsbi :G →Rof bare supposed to be in C1(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, whereGis a convex set, and for arbitrary smoothG by the results in[18]. The local timel(x)is carried by the set

C :=_{s_≥0 : X_s(x)_∈∂G_}.

We define

r(t):=sup(C∩[0,t])

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 _7→ r(t) is constant on each excursion interval of X(x)and is right-continuous. Moreover, for each t>infC we haveX_r(t)(x)_∈∂G.

2.3

Localization

In order to prove our main results we shall use the localization technique introduced in [2]. Let

{U₀,U₁, . . ._}be a countable or finite family of relatively open subsets ofG coveringG. EveryU_m is attached with a coordinate system, i.e. with a mappingu_m: U_m→Rd, giving each pointx _∈U_mthe coordinatesum(x) = (u1m(x), . . . ,udm(x))such that:

i) U_0⊆ G₀ and the corresponding coordinates are the original Euclidian coordinates. Ifm>0 the mappingu_m is one to one and twice continuously differentiable and we have

Um∩∂G={x ∈Um: u1m(x) =0}, Um∩G0={x ∈Um: u1m(x)>0}.

iii) For everym>0,_〈∇ui_m(x),n(x)_〉=δ1i for allx ∈Um∩∂G.

iv) For everym≥0 andi∈ {1, . . . ,d}, the functions

b_mi : U_m→R x_{7→ 〈∇}ui_m(x),b(x)_〉+1₂∆ui_m(x),

σi

m: Um→Rd x 7→ ∇uim(x),

satisfy

sup

m

sup

x∈Um

€

|bi_m(x)_|+_kσ_mi (x)_kŠ<∞,

and there exists a constantc, not depending onmandi, such that

|b_mi (x)₋bi_m(y)_|+_kσi_m(x)₋σi_m(y)_{k ≤}ckx−yk, ∀x,y ∈Um.

Note that these conditions implyn(x) =_∇u1_m(x)for all x _∈U_m∩∂G. Since∂G is supposed to be

C3, the functionsb_mi andσ_mi are continuously differentiable. We extend the functionsb_mi andσ_mi 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(τℓ)ℓin such a way that on each interval[τℓ,τℓ+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 δ0 ∈ (0,d0) and set ∂ˆUm := ∂Um\∂G if Um∩∂G 6= ;and

ˆ

∂Um:=∂Umif Um⊆G0. Then, we define the sequence of stopping times(τℓ)ℓby

τ0:=0, τℓ+1:=inf{t> τℓ: dist(Xt(x),∂ˆUm_ℓ)< δ0} ∧T, ℓ≥0,

where, for everyℓ≥0, m_ℓ :=m(X_τ

ℓ(x))∈Nsuch that Bd0(Xτℓ(x))⊆Umℓ. Then τℓ6∈C for every

ℓa.s. The dependence of τℓ on x will be suppressed in the notation. Note that by construction X_t(x)_∈U_mℓ for allt_∈[τℓ,τℓ+1)andmℓ6=mℓ+1 for everyℓsinceδ0<d0. For abbreviation we set

Cℓ:=C∩[τℓ,τℓ+1) ={s∈[τℓ,τℓ+1): Xs(x)∈∂G}.

Using Itô’s formula we get for t_∈[τℓ,τℓ+1):

u_mℓ(X_t(x)) =u_mℓ(X_τℓ(x)) +

Z t

τℓ

”

〈∇u_mℓ(X_r(x)),b(X_r(x))_〉+1₂∆u_mℓ(X_r(x))—d r

+

Z t

τ_ℓ

∇u_mℓ(X_r(x))d w_r+e1€l_t(x)₋l_τℓ(x)Š

=u_mℓ(Xτ_ℓ(x)) +

Z t

τ_ℓ

b_mℓ(X_r(x))d r+

Z t

τ_ℓ

σmℓ(Xr(x))d wr+e

1€

l_t(x)₋lτ_ℓ(x)

Š

. (2.2)

For everyℓwe define a continuous semimartingale(M_tx,ℓ)_t by

M_tx,ℓ:=







0 if t∈[0,τℓ),

Rt

τℓb

1

mℓ(Xr(x))d r+

Rt

τℓσ

1

mℓ(Xr(x))d wr if t∈[τℓ,τℓ+1],

M_τx,ℓ

ℓ+1 if t> τℓ+1.

Furthermore, we setL_t(x):=l_t(x)₋l_τℓ(x)if t_∈[τℓ,τℓ+1),ℓ≥0, so that

u1_m

ℓ(Xt(x)) =u

1

m_ℓ(Xτ_ℓ(x)) +Mtx,ℓ+Lt(x), t∈[τℓ,τℓ+1). (2.4)

By the Girsanov Theorem there exists a probability measure ˜Pℓ(x), which is equivalent to Pand

under whichMx,ℓ is a continuous martingale. The quadratic variation process is given by

[Mx,ℓ]_t=

Z t

τℓ

kσ1_m

ℓ(Xr(x))k

2_{d r, t}

∈[τℓ,τℓ+1),

which is strictly increasing in t on[τℓ,τℓ+1). We setρℓt :=inf{s: [M x,ℓ_]

s > t}. We can apply the

Dambis-Dubins-Schwarz Theorem, in particular its extension in Theorem V.1.7 in[23], since in our case the limit limt→∞[Mx,ℓ]t= [Mx,ℓ]τ_ℓ+1<∞exists, to conclude that the process

B_tx,ℓ:=M_ρx,ℓ

t fort<[M

x,ℓ_]

τ_ℓ+1, B

x,ℓ

t :=Mτx_ℓ,+ℓ1 fort≥[M

x,ℓ_]

τ_ℓ+1, (2.5)

is a ˜Pℓ(x)-Brownian motion w.r.t. the time-changed filtration stopped at time [Mx,ℓ]τ_ℓ+1 and we

have M_tx,ℓ = Bx,ℓ

[Mx,ℓ_] t

for all t _∈[τℓ,τℓ+1). In particular, on[τℓ,τℓ+1) the path ofMx,ℓ attains a.s. its minimum at a unique time.

Finally we introduce a moving frame. On each coordinate patchUm ofG we define a mapping x7→ O_m(x)taking values in the space of orthogonal matrices, which is twice continuously differentiable, such that for x _∈ ∂G_∩U_m the first row of O_m(x) coincides with n(x). Moreover, there exists a constantc, not depending onmsuch that

kOm(x)−Om(y)k ≤ckx−yk, ∀x,y ∈Um.

Again we extend the functionsO_m to the whole domain G such that they are uniformly Lipschitz continuous onG.

Now we define the moving frame as the right-continuous process(Ot)t∈[0,T] byOt :=Om_ℓ(Xt(x)), t_∈[τℓ,τℓ+1), which only jumps at the step timesτℓ.

We apply Itô’s formula locally on each interval[τℓ,τℓ+1)to obtain

dO_t·O_t−1=

d

X

k=1

αk(Xt(x))d wkt +β(Xt(x))d t+γ(Xt(x))d lt(x), (2.6)

for some coefficient functionsαk,βandγdepending onℓ.

2.4

Main Results

Theorem 2.1. For all t>0and x ∈G a.s. the mapping y7→Xt(y)is directional differentiable at x, i.e. the limit ηt :=ηvt(x):= DvXt(x) = limǫ→0(Xt(x+ǫv)−Xt(x))/ǫexists a.s. for every v ∈Rd. Moreover, there exists a right-continuous modification ofηsuch that a.s. for all t>0:

ηt=v+

Z t

0

D b(Xr(x))·ηrd r, if t<infC,

ηt=πXr(t)(x)(ηr(t)−) +

Z t

r(t)

D b(X_r(x))_·ηrd r, if t≥infC.

Remark 2.2. Ifx _∈∂G,t=0 is a.s. an accumulation point ofC and we haver(t)>0 a.s. for every

t>0. Therefore, in that caseη0=vandη0+=πx(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(ηt)solves (2.7), then the process(1+lt(x))ηt, 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 allf ∈C_b(G)the transition semigroupP_tf(x):=E[f(X_t(x))],x ∈G,

t>0, associated withX.

Corollary 2.4. For all f _∈C(G), t>0, x _∈G and v_∈Rd we have:

D_vP_tf(x) =1

t E



f(X_t(x)) Z t

0

d

X

k=1

〈ηv_r(x),d w_r〉



, (2.8)

and if f _∈C1(G):

DvPtf(x) =E

”

∇f(Xt(x)) ·ηvt(x)

—

. (2.9)

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 (ηt)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, whenX hits the boundary. Furthermore, ifX is at the boundary at some time t₀

and we haver(t₀₋)₆=r(t₀), i.e.t₀ is the endpoint of an excursion interval, then alsoηmay have a discontinuity at t0and jump as follows:

ηt0=πXt0(x)(ηt0−). (2.10)

Consequently, we observe that at each time t₀ as above, η is projected to the tangent space and jumps in direction of n(X_t

0(x)) or −n(Xt0(x)), respectively. Finally, if Xt0(x) ∈∂G and t 7→ r(t)

is continuous in t = t0, there is also a projection ofη, but since in this caseηt0− is in the tangent

space, the projection has no effect andηis continuous at time t₀.

Set Y_t := O_t·η_t, t ≥ 0, whereO_t denotes the moving frame introduced in Section 2.3. Let P =

diag(e1)andQ=Id₋Pand

Y_t1=P_·Y_t and Y_t2=Q_·Y_t

to decompose the spaceRd into the direct sum ImP⊕KerP. We define the coefficient functionsc(t)

andd(t)to be such that

d

X

k=1

‚

c1_k(t) c2_k(t)

c3_k(t) c4_k(t)

Œ

d wk_t +

‚

d1(t) d2(t)

d3(t) d4(t)

Œ

d t

=

d

X

k=1

αk(Xt(x))d wkt +

”

Furthermore, we setγ2(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

Y_t1=1l_{t<infCℓ} Y_τ1

ℓ+

d

X

k=1

Z t

τℓ

€

c_k1(s)Y_s1+c_k2(s)Y_s2Šd w_sk+

Z t

τℓ

€

d1(s)Y_s1+d2(s)Y_s2Šds

!

+1l_{{t≥infCℓ}}

d

X

k=1

Z t

r(t)

€

c_k1(s)Y_s1+c_k2(s)Y_s2Šd w_sk+

Z t

r(t)

€

d1(s)Y_s1+d2(s)Y_s2Šds

!

Y_t2=Y_τ2

ℓ+

d

X

k=1

Z t

τ_ℓ

€

c3_k(s)Y_s1+c_k4(s)Y_s2Šd wk_s +

Z t

τ_ℓ

€

d3(s)Y_s1+d4(s)Y_s2Šds

+

Z t

τ_ℓ

€

Φ2_s +γ2(s)ŠY_s2d ls(x),

for t∈[τℓ,τℓ+1), where

Φ2_t :=Q_·O_t·Dn(X_t(x))_·O_t−1_·Q, t_∈C =suppd l(x),

with the initial condition Y_τ1

ℓ=P·Oτℓ·O −1

τ_ℓ−·Yτ_ℓ− and Yτ2_ℓ =Q·Oτ_ℓ·Oτ−_ℓ1−·Yτ_ℓ−forℓ≥1as well as

Y₀1=P·Om0(x)·v and Y

2

0 =Q·Om0(x)·v forℓ=0.

Remark 2.6. Here and in the sequel integrals of the formR_rt_(t)ξ(s)d ws are understood as follows:

LetH: C([0,_∞))_→D([0,_∞))be the random map defined by(H f)(t):=f(t)₋f(r(t)). Then,

Z t

r(t)

ξ(s)d w_s:= (H I)(t)

whereI(t)is the continuous processI(t) =R₀tξ(s)d w_s.

The equations in Theorem 2.5 show that on every interval [τℓ,τℓ+1) the decomposition of Y is a decomposition into a discontinuous and continuous part. The discontinuous part Y1 becomes zero whenever X hits the boundary, which corresponds to the projection ofη to the tangent space described above. On the other hand,Y₂ is continuous which shows that only the component ofηin normal direction is affected by the projection.

Remark 2.7. Note that for all t∈C=suppd l(x),

Φ2_t :=Q·Ot·Dn(Xt(x))·Ot−1·Q=−Q·Ot·S(Xt(x))·O−t1·Q,

Remark 2.8. Define the process(_Mt)_t≥0, taking values inRd×d, via

ηv_t(x) =_Mt(x)_·v, v_∈Rd,t_≥0.

Then,_M is a multiplicative functional that can possibly be identified with the discontinuous multi-plicative 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 func-tional 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 forX.

Corollary 2.9. For all f _∈C(G)and t>0, the transition semigroup P_tf(x):=E[f(X_t(x))], x_∈G,

satisfies the Neumann condition at∂G:

x _∈∂G=_⇒D_n(x)P_tf(x) =0.

2.5

Example: Processes in the Unit Disc

We end this section by considering the example of the unit disc to illustrate our results. Let the domainG =B₁(0)be the closed unit disc inR2. Then, for x _∈∂G, the inner normal field is given byn(x) =₋x and the orthogonal projection onto the tangent space byπx(z) =z− 〈z,x〉x,z∈R2.

The Skorohod equation can be written as

X_t(x) = x+

Z t

0

b(X_r(x))d r+w_t−

Z t

0

X_r(x)d l_r(x), t_≥0,

X_t(x)_∈G, d l_t(x)_≥0,

Z ∞

0 1l_{kX

t(x)k<1}d lt(x) =0, t ≥0,

and the system describing the derivatives becomes

η_t=v+

Z t

0

D b(X_r(x))_·η_rd r, if t<infC,

ηt=ηr(t)−− 〈ηr(t)−,Xr(t)(x)〉Xr(t)(x) +

Z t

r(t)

D b(X_r(x))_·ηrd r, if t≥infC.

In this example a possible choice of the coordinate patches is the following. LetU0 := G0 be the interior of the disc andu₀ be the identity onU₀. Further, for some small fixedδwe set

U1:={x = (x1,x2)∈G: x1> δ}, U2:={x= (x1,x2)∈G: x1<−δ},

U3:={x = (x1,x2)∈G: x2> δ}, U4:={x= (x1,x2)∈G: x2<−δ},

and

u1_m(x):= 1

2(1− kxk

2_{), m=1, . . . , 4, u}2

m(x):=

(

arctanx2

x1 ifm=1, 2,

arctanx1

Finally, in order to define the moving frame, letO₀:=Id and

from which the coefficient functionsα1,α2andβ can be defined accordingly. Note that in any case

γ=0. Furthermore, sincen(x) =₋x for all x∈∂G, Dn(x) =₋Id and thereforeΦ2_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: Fort_∈[τℓ,τℓ+1), writingYt= (yt1,y

with initial valueYτ_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 ofx7→ (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>0we havesup_x∈Gl_t(x)<∞a.s.

Proof. See Lemma 3.3 in[4].

Proposition 3.2. Let T>0be arbitrary and let(Xt(x))and(Xt(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]k

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 oflT(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 C2-smooth and G is connected and compact, there exists a positive constantc1<∞such that for all x∈∂Gand all y∈G,

〈x₋y,n(x)_{〉 ≤}c_1kx₋y_k2. (3.1)

LetT₀:=0 and fork_≥1,

T_k:=inf¦t ≥T_k−1: kXt(x)−Xt(y)k 6∈

€₁

2kXTk−1(x)−XTk−1(y)k, 2kXTk−1(x)−XTk−1(y)k

Š©

∧T.

Then, by Itô’s formula we obtain for anyk_≥1 andt_∈(T_k−1,T_k],

kXt(x)−Xt(y)k − kXTk−1(x)−XTk−1(y)k

=

Z t

Tk−1

X_r(x)₋X_r(y),b(X_r(x))₋b(X_r(y))

kXr(x)−Xr(y)k

d r

+

Z t

Tk−1

Xr(x)−Xr(y),n(Xr(x))

kXr(x)−Xr(y)k

d l_r(x) +

Z t

Tk−1

Xr(y)−Xr(x),n(Xr(y))

kXr(x)−Xr(y)k

d l_r(y)

≤c₂

Z t

Tk−1

kX_r(x)₋X_r(y)_kd r+c₁

Z t

Tk−1

kX_r(x)₋X_r(y)_k(d l_r(x) +d l_r(y))

≤c3kXTk−1(x)−XTk−1(y)k

Z Tk

Tk−1

(d r+d lr(x) +d lr(y)),

where we have used (3.1) and the Lipschitz continuity ofb. Hence, for any t_∈(T_k−1,T_k],

kX_t(x)₋X_t(y)_k kX_Tk

−1(x)−XTk−1(y)k

≤1+c3

€

Tk−Tk−1+lTk(x)−lTk−1(x) +lTk(y)−lTk−1(y)

Š

≤exp€c₃€T_k−T_k−1+l_T

k(x)−lTk−1(x) +lTk(y)−lTk−1(y)

ŠŠ

,

and

kX_t(x)₋X_t(y)_k kx−yk =

kX_t(x)₋X_t(y)_k kX_Tk

−1(x)−XTk−1(y)k

k−1

Y

j=1

kXTj(x)−XTj(y)k

kX_Tj

−1(x)−XTj−1(y)k

≤

k

Y

j=1 exp

c₃

T_j−T_j−1+l_T

j(x)−lTj−1(x) +lTj(y)−lTj−1(y)

≤exp€c3

€

Tk+lTk(x) +lTk(y)

ŠŠ

≤exp c3 T+lT(x) +lT(y)

,

Remark 3.3. By Proposition 3.2 there exists for every T>0 a random∆_T >0 such that

sup

t∈[0,T]k

X_t(x)₋X_t(y)_k< δ0

2 , ∀y ∈B∆T(x)∩G,

withδ₀ as in Section 2.3. Then, by the definition of τℓ we have for such y and for every ℓthat X_t(y)_∈U_mℓ for allt_∈[τℓ,τℓ+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. FixT >0 andx ∈Gand set

λt(y):=

Z t

0

n(X_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 ofbthatλ(y)converges uniformly on[0,T]to

λ(x)as y tends to x.

Let now t _∈[0,T]andℓbe such that t _∈[τℓ,τℓ+1). Then, for all y ∈B∆T(x)∩G with∆T as in Remark 3.3 we have thatXs(y),Xs(x)∈Um_ℓ for alls∈[τℓ,t). For such y andswe get

d ls(y)−d ls(x) =〈∇u1mℓ(Xs(y)),n(Xs(y))〉d ls(y)− 〈∇u

1

mℓ(Xs(x)),n(Xs(x))〉d ls(x) =_∇u1_m

ℓ(Xs(y))dλs(y)− ∇u

1

m_ℓ(Xs(x))dλs(x)

=σ1_m

ℓ(Xs(y))dλs(y)−σ

1

mℓ(Xs(x))dλs(x).

Hence,

l_t(y)₋l_t(x) =l_τ

ℓ(y)−lτℓ(x) +

Z t

τℓ

σ1_m

ℓ(Xs(x)) (dλs(y)−dλs(x))

+

Z t

τℓ

σ1_m

ℓ(Xs(y))−σ

1

mℓ(Xs(x))

dλs(y).

Using Proposition 3.2 and the fact that the functionsσm are uniformly Lipschitz the last term

con-verges to zero as y tends to x. Recall thatλ(y)converges uniformly on[τℓ,t]toλ(x)as y tends

to x. Hence, we have that the associated signed measures dλ(y) on [τℓ,t] converge weakly to

dλ(x)as ytends tox. Sinces_7→σ1_m

ℓ(Xs(x))is bounded and continuous on[τℓ,t], the second term

converges to zero as y tends tox. We apply the same argument forlτ_ℓ(y)−lτ_ℓ(x)on[τℓ−1,τℓ]and

by iterating this procedure we obtain the claim.

We fix now an arbitrary x _∈G, v_∈Rd andT >0. Then, we set x_ǫ:= x+ǫv for allǫ∈[a_x,b_x]

withax ≤0 and bx ≥0 such thatxǫ∈Gfor allǫ∈[ax,bx]. Furthermore, we define for suchǫand t_∈[0,T],

M_tx,ℓ(ǫ):=







0 if t_∈[0,τℓ),

Rt

τ_ℓb

1

mℓ(Xr(xǫ))d r+

Rt

τ_ℓσ

1

mℓ(Xr(xǫ))d wr if t∈[τℓ,τℓ+1],

M_τx,ℓ

The index x is there to indicate that the stopping timesτℓ are the same as in the definition of Mx,ℓ

that are depending on x and not on ǫ. In particular, Mx,ℓ(ǫ) is a well-defined object, since the coefficient functions b1_m and σ1

m have been extended to the whole domain G. Note that M x,ℓ

t =

M_tx,ℓ(0),t_∈[0,T]. Finally, we set

∆M_tx,ℓ(ǫ,ǫ′):=M_tx,ℓ(ǫ)₋M_tx,ℓ(ǫ′), t_∈[0,T],ǫ,ǫ′∈[a_x,b_x],

so that

∆M_tx,ℓ(ǫ, 0) =

Z t

τ_ℓ

b1_m

ℓ(Xr(xǫ))−b

1

mℓ(Xr(x))

d r+

Z t

τ_ℓ

σ1_m

ℓ(Xr(xǫ))−σ

1

mℓ(Xr(x))

d w_r,

for t _∈[τℓ,τℓ+1) and ǫ ∈[ax,bx]. In the next lemma we show that M x,ℓ

t (ǫ) is pathwise jointly

continuous int andǫ.

Lemma 3.5.Let∆_T be as in Remark 3.3. Then, for a.e.ω∈Ωthe following holds. For everyδ1∈(0, 1)

andδ2∈(0,1₂)there exists a random constant K=K(ω,δ1,δ2,T)such that

∆M

x,ℓ

t (ǫ,ǫ′)−∆Mx, ℓ s (ǫ,ǫ′)

≤K|ǫ−ǫ

′_|1−δ1_|t−s|12−δ2_{, ∀s,t∈[0,T], (3.2)}

for allǫ,ǫ′such that x_ǫ,x_ǫ′∈B_∆

T(x)∩G. In particular, for everyδ∈(0, 1)we have for all suchǫ

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(Mℓ,x(ǫ))_t,ǫsatisfying 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 b1_m andσ1_mand

the Burkholder inequality that for everyp>1 there exists a positive constantc1=c1(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 functionsb1_mandσ1

mare uniformly bounded and again by using Burkholder’s

inequal-ity we get that for everyp>1

E

• M

x,ℓ

t (ǫ)−Msx,ℓ(ǫ)

p˜

≤c_2|t₋s_|p/2, _∀s,t_∈[0,T],ǫ∈[a_x,b_x]

for some constant c₂ = c₂(p,T). Next we will show that for every p > 1 there exists a constant

c3=c3(p,T)such that

E

• ∆M

x,ℓ

t (ǫ,ǫ′)−∆Msx,ℓ(ǫ,ǫ′)

p˜

≤c_3|ǫ−ǫ′|p|t₋s_|p/2, _∀s,t_∈[0,T],ǫ,ǫ′∈[a_x,b_x].

Mx,ℓ(ǫ′) are defined to be constant on [0,T]_\[τℓ,τℓ+1]. Thus, it is enough to consider the case

By the uniform Lipschitz continuity of bmand Proposition 3.2 the first term can be estimated by

c_|t₋s_|pE

For the second term we get the following estimate by Burkholder’s inequality, the uniform Lipschitz continuity ofσm and again by Proposition 3.2:

cE

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 δ1 ∈(0, 1) andδ2 ∈(0,₂1) there exists a modification of the random field (Mx,ℓ(ǫ))t,ǫ

satisfying (3.2) for some random constantK=K(ω,δ1,δ2,T).

Step 2: The existence of a modification shown in Step 1 immediately implies that a.s. (3.2) holds

for alls,t_∈[0,T]_∩Qandǫ,ǫ′∈[a_x,b_x]_∩Q. The claim follows if∆M_tx,ℓ(ǫ,ǫ′)₋∆M_sx,ℓ(ǫ,ǫ′) is pathwise continuous insandt as well as inǫandǫ′. It is enough to show the continuity of M_tx,ℓ(ǫ)

in t andǫ. The continuity in t is obvious and for everyǫsuch that x_ǫ∈B_∆

T(x)∩G we get by an application of Itô’s formula as in (2.2)

M_tx,ℓ(ǫ) =u1_m

ℓ(Xt(xǫ))−u

1

mℓ(Xτℓ(xǫ))−Lt(xǫ),

where the right hand side is continuous inǫby Proposition 3.2 and Lemma 3.4.

3.2

Convergence of Minimum Times

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 martingaleMx,ℓattains its minimum. As a result we shall obtain that forǫtending to zero the minimum time ofMx,ℓconverges almost surely faster than polynomially to the minimum time ofMx,ℓ.

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, An is open and recall that for everyℓ the mapping t 7→ [Mx,ℓ]t

is continuous and increasing on [τℓ,τℓ+1). Thus, for every n we may choose a random qn ∈An

a follows: Let ℓ be such that infA_{n ∈} [τℓ,τℓ+1), then we choose qn ∈ [τℓ,τℓ+1)∩An such that

[Mx,ℓ]_q n∈Q.

In order to compute the local timel(x), recall that on every interval[τℓ,τℓ+1),ℓ≥0,l(x)is carried by the set of times t, when u1_m

ℓ(Xt(x)) =0. Therefore, we can apply Skorohod’s Lemma (see e.g.

Lemma VI.2.1 in[23]) to equation (2.4) to obtain

Lt(x) =

−u1_m

ℓ(Xτℓ(x))−_τinf ℓ≤s≤t

M_sx,ℓ

+

, t∈[τℓ,τℓ+1).

Fix anyqn > infC and ℓ such that qn ∈ [τℓ,τℓ+1) . Since u1mℓ(Xr(qn)(x)) = 0 and t 7→ Lt(x) is non-decreasing, we have for allτℓ≤s≤r(qn):

M_rx₍,_qℓ

n)=−u 1

m_ℓ(Xτ_ℓ(x))−Lr(qn)(x)≤ −u 1

m_ℓ(Xτ_ℓ(x))−Ls(x)

=₋u1_m

ℓ(Xs(x)) +M

x,ℓ

s ≤M

x,ℓ

s .

Moreover, L(x)is constant on[r(qn),t]for allt∈An∩[τℓ,τℓ+1), so that

L_t(x) =L_r(qn)(x) =

h

−u1_m

ℓ(Xτℓ(x))−M

x,ℓ r(qn)

i+

, t ∈A_n∩[τℓ,τℓ+1). (3.3)

Note thatM_rx₍,_qℓ

n)≤ M

x,ℓ

s for alls∈[τℓ,qn]. Further, with probability one we have that r(qn)is the

unique time in[τℓ,qn], whenMx,ℓ attains its minimum. Analogously we compute the local time of

the process with perturbed starting point. For fixed v∈Rd we set xǫ:=x+ǫv,ǫ∈R, where|ǫ|is

always supposed to be sufficiently small, such that xǫlies inG. Furthermore, there exists a random

∆_n >0 such that for allǫ∈(₋∆_n,∆_n)we haveX_t(x_ǫ)_∈U_mℓ for all t _∈[τℓ,qn](cf. Remark 3.3).

As above we obtain for suchǫ:

L_qn(xǫ) =Lrǫ(qn)(xǫ) =

h

−u1_m

ℓ(Xτℓ(xǫ))−M

x,ℓ rǫ(qn)(ǫ)

i+

, (3.4)

wherer_ǫ(q_n)is defined similarly asr(q_n). In particular,M_rx,ℓ

ǫ(qn)(ǫ)≤M

x,ℓ

s (ǫ)for alls∈[τℓ,qn].

Lemma 3.6. For all n we have r_ǫ(q_n)_→r(q_n)a.s. forǫ→0.

Proof. Consider someqn and letℓbe such that qn ∈[τℓ,τℓ+1). We fix now a typical ωsuch that

r(q_n)is the unique time in[τℓ,q_n], whenMx,ℓattains its minimum and such that Lemma 3.5 holds. For every sequence(ǫk)k converging to zero we can extract a subsequence of(rǫk(qn)), still denoted by(rǫk(qn)), converging to someˆr(qn). By construction we have

M_rx,ℓ

ǫk(qn)

for everyk. Note that on one hand the right hand side converges toM_rx₍,_qℓ

n)ask→ ∞by Lemma 3.5. On the other hand the left hand side converges toM_ˆrx₍,_qℓ

n), since

M_rx,ℓ

ǫk(qn)

(ǫk)−M x,ℓ

ˆ

r(qn)

≤

M_rx,ℓ

ǫk(qn)

(ǫk)−M x,ℓ rǫk(qn)

+

M_rx,ℓ

ǫk(qn)−

M_ˆrx₍,_qℓ

n)

,

where the first term tends to zero fork→ ∞by Lemma 3.5 and the second term by the continuity of

Mx,ℓ. Thus,M_ˆrx₍,_qℓ

n)≤M

x,ℓ

r(qn). Sincer(qn)is unique time in[τℓ,qn], when M

x,ℓ _{attains its minimum,}

this impliesˆr(qn) =r(qn).

Lemma 3.7. Let(Wt)t≥0 be a Brownian motion on(Ω,F,P). For all T >0, letϑ:Ω→[0,T]be the

random variable such that a.s.

W_ϑ<W_{s ∀}s_∈[0,T]_\{ϑ}.

Then,

lim inf

s→ϑ

W_s−W_ϑ

p

|s₋ϑ|h(_|s₋ϑ|) ≥

1 a.s., (3.5)

for every function h on[0,_∞)satisfying0<h(t)_↓0as t_↓0andRr0

0 h(t)

d t

t <∞for some r0>0.

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,Mˆ) two independent copies of the standard Brownian meander (see[23]), we set for allr_∈(0, 1),

V_r(t):=

(

−pr M(1) +pr M(r−_rt), t_∈[0,r] −pr M(1) +p1₋rMˆ(t−r

1−r), t∈(r, 1]

Let now(τ,M,Mˆ) be an independent triple, such thatτhas the arcsine law. Then,V_τ =d W. This formula has the following meaning: τis the unique time in[0, 1], when the path attains minimum

−pτM(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 of Mˆ. 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 (Rt)t≥0 on the time interval[0, 1]starting in zero. We recall that a.s.

lim inf

t→0

R_t

p

t h(t) ≥1

for every function hsatisfying 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 motionsBx,ℓdefined in Section 2.3. More precisely, Lemma 3.7 gives that a.s. for every ℓand every nonnegative q _∈Qthe following holds: Ifq≤[Mx,ℓ]_τ

ℓ+1, denoting byϑ

ℓ

qthe unique time whenBx

,ℓ_{attains its minimum over[0,q],}

Proposition 3.8. Letδ >0be arbitrary. Then, for all n we have

|rǫ(qn)−r(qn)|δ

ǫ −→0 a.s. asǫ→0.

Proof. First we fix 0< δ′<1. By construction we have for everyqn andℓsuch thatqn∈[τℓ,τℓ+1) and forǫsmall enough,

M_rx,ℓ

and we derive from (3.6) that a.s.

c₁r_ǫ(q_n)−r(q_n)

Corollary 3.9. For any n and letℓbe such that q_n∈[τℓ,τℓ+1). Then, continuous of order 1₂−δ, we obtain i) from Proposition 3.8.

ii) follows from i). Indeed, by Proposition 3.2 and Lemma 3.6 we have forǫsufficiently small that

lr(qn)(xǫ) =lrǫ(qn)(xǫ) =0 ifqn<infC andlr(qn)(xǫ),lrǫ(qn)(xǫ)>0 ifqn>infC. In the first case ii) is trivial and the latter case we have by (3.4)

3.3

Proof of the Differentiability

exists and is measurable. Furthermore, for allω∈Ω₀,( ˆηt)t∈[0,T]\C satisfies

ˆ

ηt=v+

Z t

0

D b(Xr(x))·ηˆkrd r, t∈[0, infC),

ˆ

ηt= ˆηr(t)+

Z t

r(t)

D b(X_r(x))_·ηˆk_rd r, t_∈[infC,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 thebeginningof the proof of Proposition 3.10 we will choose the set

Ω₀ with full measure. No sequence or subsequence is involved in this choice. The statement of the Proposition will then follow by completelypathwisearguments, which arepurely deterministicand do not require any other choice of events.

In the next step we extendηˆ(ω),ω∈Ω₀, to a right-continuous trajectory on[0,T]. Then, we prove that in coordinates of the moving frame introduced in Section 2.2ηˆsolves 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 thatηT(ǫ)converges a.s. if and only if for every componentηiT(ǫ),i=1, . . . ,d,

lim inf

ǫ→0 η

i

T(ǫ) =lim sup ǫ→0

ηi_T(ǫ) a.s. (3.9)

Let now(ǫ_νi,−)ν and (ǫνi,+)ν be two random sequences, along which ηiT(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 ηˆ− andηˆ+, being two trajectories in Rd whose i-th components at time

T, ηˆi_T,− andηˆ_Ti,+, coincide with the limes inferior and limes inferior of ηi_T(ǫ), of course. From the fact that both ηˆ− andηˆ+ solve an equation having a pathwise unique solution we conclude that

P

h

ˆ

η−_t = ˆη+_t,_∀t_∈[0,T]

i

=1, which impliesηˆi_T,−= ˆηi_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 chooseΩ_{0 ⊆} Ω with full measure such that

for allω ∈Ω₀ Lemma 3.1 holds and Corollary 3.9 holds for all n. Let nowω ∈Ω₀ be fixed. Let

t _∈[0,T]_\C andnbe such that t _∈A_n. Using Proposition 3.2 there exists ∆_n = ∆_n(ω)>0 such that l_q

n(x) = lqn(xǫ) =0 if qn < infC and both of them are strictly positive if qn > infC for all

|ǫ| ∈(0,∆_n). Then,

ηt(ǫ) =ηr(qn)(ǫ) + 1

ǫ

Z t

r(qn)

b(X_r(x_ǫ))₋b(X_r(x))

d r+1

ǫ

Z r_ǫ(q_n)

0

n(X_r(x_ǫ))d l_r(x_ǫ)

−1

ǫ

Z r(q_n)

0

n(X_r(x_ǫ))d l_r(x_ǫ)

=ηr(qn)(ǫ) +

d

X

k=1

Z t

r(qn)



 Z 1

0

∂b ∂xk(X

α,ǫ r )dα



ηk_r(ǫ)d r+R_q

whereXα_r,ǫ:=αXr(xǫ) + (1−α)Xr(x),α∈[0, 1], and

R_qn(xǫ):=

1

ǫ

Z r_ǫ(q_n)

r(qn)

n(X_r(xǫ))d lr(xǫ). (3.11)

Note that ifq_n<infC we haver(q_n) =0,ηr(qn)(ǫ) =vandRqn(xǫ) =0. In any case,

kR_q

n(xǫ)k ≤ 1

ǫ

Z r_ǫ(q_n)

r(qn)

kn(X_r(x_ǫ))_kd l_s(x_ǫ)

=

l_rǫ(q

n)(xǫ)−lr(qn)(xǫ)

ǫ

−→

0 asǫ→0, (3.12)

by Corollary 3.9. Recall thatkηt(ǫ)k ≤exp(c1(T+lT(x) +lT(xǫ)))≤c for allt ∈[0,T]andǫ6=0

by Proposition 3.2 and Lemma 3.1. Let(ǫν)ν = (ǫν(ω))ν be any random sequence converging to

zero. By a diagonal procedure, we can extract a subsequence(ǫνl)l with(νl)l = (νl(ω))l such that

ηr(qn)(ǫνl)has a limitηˆr(qn)∈R

d _{asl→ ∞for alln∈N.}

Let nowηˆ: [0,T]_\C _→Rd be the unique solution of

ˆ

η_t:= ˆη_r(qn)+

Z t

r(qn)

D b(X_r(x))_·ηˆ_rd r, t ∈A_n.

By (3.10) and Proposition 3.2, we get for_|ǫ| ∈(0,∆_n)andt_∈A_n,

kηt(ǫ)−ηˆtk ≤kηr(qn)(ǫ)−ηˆr(qn)k+kRqn(xǫ)k +sup

r∈An

D b(Xα,ǫ

r )−D b(Xr(x))

exp(c₁(T+l_T(x) +l_T(x_ǫ)))

+c₂

Z t

0

kηr(ǫ)−ηˆrkd r.

Since ηr(qn)(ǫνl) →ηˆr(qn), kRqn(xǫ)k → 0 , X

α,ǫν_l

r → Xr(x) uniformly in r ∈ [0,t] and since the

derivatives of b are continuous, we obtain by Gronwall’s Lemma thatη_t(ǫνl) converges toηˆt uni-formly in t_∈A_nfor every n. Thus, sinceC has zero Lebesgue measure,ηt(ǫνl)converges toηˆt for all t∈[0,T]_\C asl→ ∞and by the dominated convergence theorem we get that( ˆηt(ω))t∈[0,T]\C

satisfies (3.8). Finally, the measurability ofηˆis immediate from its construction.

From now on we will denote by ηˆ the limiting object constructed in Proposition 3.10 from any arbitrary but fixed random sequence (ǫn). The next lemma shows that ηˆr(qn) is in the tangent at

X_r(q

n)(x)for everyqn.

Lemma 3.11. For every q_n>infC,

i) _〈ηr(qn)(ǫ),n(Xr(qn)(x))〉 →0a.s. and in L

p_{, p>1, asǫ}

→0,

ii) 〈ηr_ǫ(qn)(ǫ),n(Xrǫ(qn)(xǫ))〉 →0a.s. and in L

p_{, p>1, asǫ}

→0.

Proof. By dominated convergence it suffices to prove convergence almost surely. Letℓbe such that

qn∈[τℓ,τℓ+1). Then, clearlyXr(qn)(x)∈Umℓ∩∂G. Recall thatn(Xr(qn)(x)) =∇u 1

m_ℓ(Xr(qn)(x)), and by Taylor’s formula we get

〈ηr(qn)(ǫ),n(Xr(qn)(x))〉= 1

ǫ

u1_m

ℓ(Xr(qn)(xǫ))−u 1

mℓ(Xr(qn)(x))

Note that the term of second order in the Taylor expansion is inO(ǫ)by Proposition 3.2. Recall that formula and the fact thatu1_m

ℓ(Xrǫ(qn)(xǫ)) =0,

which tends to zero again by Corollary 3.9 i).

Now we define fort_∈C_∩[τℓ,τℓ+1)

Proof. By Proposition 3.2 and Lemma 3.1 the first term can be estimated as follows

E

For the second term we get similarly, using Burkholder’s inequality,

E

Hence both terms converges to zero along(ǫνk) by dominated convergence, since X

Lemma 3.13. For almost everyωthe following holds.

In particular, the trajectories ofηˆare right-continuous on[0,T].