Differentiability of stochastic flow of reflected Brownian motions
aa r X i v : . [ m a t h . P R ] J un DIFFERENTIABILITY OF STOCHASTIC FLOW OF REFLECTEDBROWNIAN MOTIONS
KRZYSZTOF BURDZY
Abstract.
We prove that a stochastic flow of reflected Brownian motions in a smoothmultidimensional domain is differentiable with respect to its initial position. The derivativeis a linear map represented by a multiplicative functional for reflected Brownian motion. Themethod of proof is based on excursion theory and analysis of the deterministic Skorokhodequation. Introduction
This article contains a result on a stochastic flow X xt of reflected Brownian motions in asmooth bounded domain D ⊂ R n , n ≥
2. We will prove that for some stopping times σ r defined later in the introduction, the mapping x → X xσ r is differentiable a.s., and we willidentify the derivative with a mapping already known in the literature.We start with an informal overview of our research project. We call a pair of reflectedBrownian motions X t and Y t in D a synchronous coupling if they are both driven by thesame Brownian motion. To make things interesting, we assume that X = Y . The ultimategoal of the research project of which this paper is a part, is to understand the long timebehavior of V t := X t − Y t in smooth domains. This project was started in [BCJ], where syn-chronous couplings in 2-dimensional smooth domains were analyzed. An even earlier paper[BC] was devoted to synchronous couplings in some classes of planar non-smooth domains.Multidimensional domains present new challenges due to the fact that the curvature of ∂D is not a scalar quantity and it has a significant influence on V t . Eventually, we would liketo be able to prove a theorem analogous to the main result of [BCJ], Theorem 1.2. Thattheorem shows that | V t | goes to 0 exponentially fast as t goes to infinity, provided a certainparameter Λ( D ) characterizing the domain D is strictly positive. The exponential rate atwhich | V t | goes to 0 is equal to Λ( D ). The proof of Theorem 1.2 in [BCJ] is extremely longand we expect that an analogous result in higher dimensions will not be easier to prove.This article and its predecessor [BL] are devoted to results providing technical backgroundfor the multidimensional analogue of Theorem 1.2 in [BCJ]. Mathematics Subject Classification.
Key words and phrases.
Reflected Brownian motion, multiplicative functional.Research supported in part by NSF Grant DMS-0600206.
Suppose that | V t | is very small for a very long time. Then we can think about the evolutionof V t as the evolution of an infinitesimally small vector, or a differential form, associated to X t . This idea is not new—in fact it appeared in somewhat different but essentially equivalentways in [A, IK1, IK2, H]. The main theorem of [BL] showed existence of a multiplicativefunctional governing the evolution of V t , using semi-discrete approximations. The result doesnot seem to be known in this form, although it is close to theorems in [A, IK1, H]. However,the main point of [BL] was not to give a new proof to a slightly different version of a knownresult but to develop estimates using excursion techniques that are analogous to those in[BCJ], and that can be applied to study V t .Suppose that for every x ∈ D we have a reflecting Brownian motion X xt in D startingfrom X x = x , and all processes X xt , x ∈ D , are driven by the same Brownian motion. For afixed x ∈ D , let σ r be the first time t when the local time of X x on ∂D reaches the value r .The main result of the present article, Theorem 3.1, says that for every r >
0, the mapping x → X xσ r is differentiable at x = x a.s., and the derivative is a linear mapping defined inTheorem 3.2 of [BL].The differentiability in the initial data was proved in [DZ] for a stochastic flow of reflecteddiffusions. The main difference between our result and that in [DZ] is that that paper wasconcerned with diffusions in (0 , ∞ ) n , and our main goal is to study the effect of the curvatureof ∂D . The results in [DZ] have been transferred to SDEs in a convex polyhedron with pos-sibly oblique reflection—see [An]. Differentiability of a stochastic flow of diffusions (withoutreflection) in the initial condition is a classical topic, see, e.g., [K], Chap. II, Thm. 3.1.Our main result can be considered a pathwise version of theorems proved in [A, H, IK1]and [IK2], Section V.6 (see also references therein). In a sense, we pass to the limit ina different order than the authors of the cited publications. Hence, our theorem is closerin spirit to the results in [LS, S, DI, DR]. There is a difference, though. The articles[LS, S, DI, DR] are concerned with the transformation of the whole driving path into areflected path (the “Skorokhod map”). At this level of generality, the Skorokhod map wasproved to be H¨older with exponent 1/2 in Theorems 1.1 an 2.2 of [LS] and Lipschitz inProposition 4.1 in [S]. See [S] for further references and history of the problem. Undersome other assumptions, the Skorokhod map was proved to have the Lipschitz property in[DI, DR]. Articles [MM] (Lemma 5.2) and [MR] contain results about directional derivativesof the Skorokhod map in an orthant, without and with oblique reflection, respectively. Thefirst theorems on existence and uniqueness of solutions to the stochastic differential equationrepresenting reflected Brownian motion were given in [T]. Some results on stochastic flows ofreflected Brownian motions were proved in an unpublished thesis [W]. Synchronous couplings tochastic flow of reflected Brownian motions 3 in convex domains were studied in [CLJ1, CLJ2], where it was proved that under mildassumptions, V t is not 0 at any finite time.The proof of the main result depends in a crucial way on ideas developed in a joint projectwith Jack Lee ([BL]). I am indebted to him for his implicit contributions to this paper. Iam grateful to Sebastian Andres, Peter Baxendale, Elton Hsu and Kavita Ramanan for veryhelpful advice. 2. Preliminaries
General notation.
All constants are assumed to be strictly positive and finite, unlessstated otherwise. The open ball in R n with center x and radius r will be denoted B ( x, r ).We will use d ( · , · ) to denote the distance between a point and a set.2.2. Differential geometry.
We will review some notation and results from [BL]. Wewill be concerned with a bounded domain D ⊂ R n , n ≥
2, with a C boundary ∂D . Wemay consider M := ∂D to be a smooth, properly embedded, orientable hypersurface (i.e.,submanifold of codimension 1) in R n , endowed with a smooth unit normal vector field n .We consider M as a Riemannian manifold with the induced metric. We use the notation h · , · i for both the Euclidean inner product on R n and its restriction to the tangent space T x M for any x ∈ M , and | · | for the associated norm. For any x ∈ M , let π x : R n → T x M denote the orthogonal projection onto the tangent space T x M , so π x z = z − h z , n ( x ) i n ( x ) , (2.1)and let S ( x ) : T x M → T x M denote the shape operator (also known as the Weingarten map ),which is the symmetric linear endomorphism of T x M associated with the second fundamentalform. It is characterized by S ( x ) v = − ∂ v n ( x ) , v ∈ T x M, (2.2)where ∂ v denotes the ordinary Euclidean directional derivative in the direction of v . If γ : [0 , T ] → M is a smooth curve in M , a vector field along γ is a smooth map v : [0 , T ] → M such that v ( t ) ∈ T γ ( t ) M for each t . The covariant derivative of v along γ is given by D t v ( t ) := v ′ ( t ) − h v ( t ) , S ( γ ( t )) γ ′ ( t ) i n ( γ ( t ))= v ′ ( t ) + h v ( t ) , ∂ t ( n ◦ γ )( t ) i n ( γ ( t )) . The eigenvalues of S ( x ) are the principal curvatures of M at x , and its determinant is theGaussian curvature. We extend S ( x ) to an endomorphism of R n by defining S ( x ) n ( x ) = 0.It is easy to check that S ( x ) and π x commute, by evaluating separately on n ( x ) and on v ∈ T x M .For any linear map A : R n → R n , we let kAk denote the operator norm. tochastic flow of reflected Brownian motions 4 We recall two lemmas from [BL].
Lemma 2.1.
For any bounded C domain D ⊂ R n and c , there exists c such that thefollowing estimates hold for all x, y ∈ ∂D , ≤ l, r ≤ c , b ≥ and z ∈ R n : k e b S ( x ) k ≤ e c b . (2.3) k e l S ( x ) − Id k T x ≤ c l. (2.4) k e l S ( x ) − e l S ( y ) k ≤ c l | x − y | . (2.5) k e l S ( x ) − e r S ( x ) k ≤ c | l − r | . (2.6) | n ( x ) − n ( y ) | ≤ c | x − y | . (2.7) Lemma 2.2.
For any bounded C domain D ⊂ R n , there exists a constant c such that forall w, x, y, z ∈ ∂D , the following operator-norm estimate holds: k π z ◦ ( π y − π x ) ◦ π w k ≤ c ( | w − y | | y − z | + | w − x | | x − z | ) . Remark 2.3.
Since ∂D is C , it is elementary to see that there exist r > ν ∈ (1 , ∞ )with the following properties. For all x, y ∈ ∂D , z ∈ D , with | x − y | ≤ r and | x − z | ≤ r ,1 − ν | x − y | ≤ h n ( x ) , n ( y ) i ≤ , (2.8) |h x − y, n ( x ) i| ≤ ν | x − y | , (2.9) h x − z, n ( x ) i ≤ ν | x − z | , (2.10) h x − z, n ( y ) i ≤ ν | x − y | | x − z | , (2.11) | π y ( n ( x )) | ≤ ν | x − y | . (2.12)If x, y ∈ ∂D , z ∈ D and | π x ( z − y ) | ≤ | π x ( x − y ) | ≤ r then h z − y, n ( x ) i ≥ − ν | π x ( x − y ) | | π x ( z − y ) | . (2.13)2.3. Probability.
Recall that D ⊂ R n , n ≥
2, is an open connected bounded set with C boundary and n ( x ) denotes the unit inward normal vector at x ∈ ∂D . Let B be standard d -dimensional Brownian motion and consider the following Skorokhod equation, X xt = x + B t + Z t n ( X xs ) dL xs , for t ≥ . (2.14)Here x ∈ D and L x is the local time of X x on ∂D . In other words, L x is a non-decreasingcontinuous process which does not increase when X x is in D , i.e., R ∞ D ( X xt ) dL xt = 0, a.s.Equation (2.14) has a unique pathwise solution ( X x , L x ) such that X xt ∈ D for all t ≥ X x is a strong Markov process. The results in [LS]are deterministic in nature, so with probability 1, for all x ∈ D simultaneously, (2.14) has aunique pathwise solution ( X x , L x ). In other words, there exists a stochastic flow ( x, t ) → X xt ,in which all reflected Brownian motions X x are driven by the same Brownian motion B . tochastic flow of reflected Brownian motions 5 We fix a point z ∈ D . We will abbreviate ( X z , L z ) by writing ( X, L ).We need an extra “cemetery point” ∆ outside R n , so that we can send processes killedat a finite time to ∆. For s ≥ X s ∈ ∂D we let ζ ( e s ) = inf { t > X s + t ∈ ∂D } .Here e s is an excursion starting at time s , i.e., e s = { e s ( t ) = X t + s , t ∈ [0 , ζ ( e s )) } . We let e s ( t ) = ∆ for t ≥ ζ ( e s ), so e t ≡ ∆ if ζ ( e s ) = 0.Let σ be the inverse of local time L , i.e., σ t = inf { s ≥ L s ≥ t } , and E r = { e s : s <σ r } . Fix some r, ε > { e u , e u , . . . , e u m } be the set of all excursions e ∈ E r with | e (0) − e ( ζ − ) | ≥ ε . We assume that excursions are labeled so that u k < u k +1 for all k and welet ℓ k = L u k for k = 1 , . . . , m . We also let u = inf { t ≥ X t ∈ ∂D } , ℓ = 0, ℓ m +1 = r , and∆ ℓ k = ℓ k +1 − ℓ k . Let x k = e u k ( ζ − ) be the right endpoint of excursion e u k for k = 1 , . . . , m ,and x = X u .Recall from Section 2.2 that S denotes the shape operator and π x is the orthogonal pro-jection on the tangent space T x ∂D , for x ∈ ∂D . For v ∈ R n , let v r = exp(∆ ℓ m S ( x m )) π x m · · · exp(∆ ℓ S ( x )) π x exp(∆ ℓ S ( x )) π x v . (2.15)Note that all concepts based on excursions e u k depend implicitly on ε >
0, which is oftensuppressed in the notation. Let A εr denote the linear mapping v → v r .We will impose a geometric condition on ∂D . To explain its significance, we consider D such that ∂D contains n non-degenerate ( n − { X t , ≤ t ≤ r } visits the n balls and no other part of ∂D , then it is easy to see that A εr = 0. To avoid thisuninteresting situation, we impose the following assumption on D . Assumption 2.4.
For every x ∈ ∂D , the ( n − -dimensional surface area measure of { y ∈ ∂D : h n ( y ) , n ( x ) i = 0 } is zero. The following theorem has been proved in [BL].
Theorem 2.5.
Suppose that D satisfies all assumption listed so far in Section 2. Then forevery r > , a.s., the limit A r := lim ε → A εr exists and it is a linear mapping of rank n − .For any v , with probability 1, A εr v → A r v as ε → , uniformly in r on compact sets. Let t = inf { t ≥ X t ∈ ∂D } and z = X t . Intuitively speaking, A r is defined by v ( r ) = A r v , where v ( t ) represents the solution to the following ODE, D v = ( S ◦ X ( σ t )) v dt, v (0) = π z v . In the 2-dimensional case, and only in the 2-dimensional case, we have an alternativeintuitive representation of |A r v | . If v = ( v , v ) then we write b v = ( − v , v ). Let µ ( x ) be tochastic flow of reflected Brownian motions 6 the curvature at x ∈ ∂D , that is, the eigenvalue of S ( x ). Then |A r v | = exp (cid:18)Z r µ ( X σ t ) dL t (cid:19) |h n ( z ) , b v i| Y e s ∈E r |h n ( e s (0)) , n ( e s ( ζ − )) i| . The remaining part of this section is a short review of the excursion theory. See, e.g., [M]for the foundations of the excursion theory in the abstract setting and [Bu] for the specialcase of excursions of Brownian motion. Although [Bu] does not discuss reflected Brownianmotion, all results we need from that book readily apply in the present context.An “exit system” for excursions of the reflected Brownian motion X from ∂D is a pair( L ∗ t , H x ) consisting of a positive continuous additive functional L ∗ t and a family of “excursionlaws” { H x } x ∈ ∂D . In fact, L ∗ t = L t ; see, e.g., [BCJ]. Recall that ∆ denotes the “cemetery”point outside R n and let C be the space of all functions f : [0 , ∞ ) → R n ∪ { ∆ } which arecontinuous and take values in R n on some interval [0 , ζ ), and are equal to ∆ on [ ζ , ∞ ).For x ∈ ∂D , the excursion law H x is a σ -finite (positive) measure on C , such that thecanonical process is strong Markov on ( t , ∞ ), for every t >
0, with transition probabilitiesof Brownian motion killed upon hitting ∂D . Moreover, H x gives zero mass to paths which donot start from x . We will be concerned only with “standard” excursion laws; see Definition3.2 of [Bu]. For every x ∈ ∂D there exists a unique standard excursion law H x in D , up toa multiplicative constant.Recall that excursions of X from ∂D are denoted e s and σ t = inf { s ≥ L s ≥ t } . Let I be the set of left endpoints of all connected components of (0 , ∞ ) r { t ≥ X t ∈ ∂D } . Thefollowing is a special case of the exit system formula of [M], E "X t ∈ I W t · f ( e t ) = E Z ∞ W σ s H X ( σ s ) ( f ) ds = E Z ∞ W t H X t ( f ) dL t , (2.16)where W t is a predictable process and f : C → [0 , ∞ ) is a universally measurable functionwhich vanishes on excursions e t identically equal to ∆. Here H x ( f ) = R C f dH x .The normalization of the exit system is somewhat arbitrary, for example, if ( L t , H x ) isan exit system and c ∈ (0 , ∞ ) is a constant then ( cL t , (1 /c ) H x ) is also an exit system. Let P yD denote the distribution of Brownian motion starting from y and killed upon exiting D .Theorem 7.2 of [Bu] shows how to choose a “canonical” exit system; that theorem is statedfor the usual planar Brownian motion but it is easy to check that both the statement andthe proof apply to the reflected Brownian motion in R n . According to that result, we cantake L t to be the continuous additive functional whose Revuz measure is a constant multipleof the surface area measure on ∂D and H x ’s to be standard excursion laws normalized sothat H x ( A ) = lim δ ↓ δ P x + δ n ( x ) D ( A ) , (2.17) tochastic flow of reflected Brownian motions 7 for any event A in a σ -field generated by the process on an interval [ t , ∞ ), for any t > L is the measure dx/ (2 | D | ) on ∂D , i.e., if the initial distribution of X is the uniform probability measure µ in D then E µ R A ( X s ) dL s = R A dx/ (2 | D | ) for anyBorel set A ⊂ ∂D , see Example 5.2.2 of [FOT]. It has been shown in [BCJ] that ( L t , H x ) isan exit system for X in D , assuming the above normalization.3. Differentiability of the stochastic flow in the initial parameter
Recall that z ∈ D is a fixed point. Our main result is the following theorem. Theorem 3.1.
Suppose that D satisfies all assumptions of Section 2. Then for every r > and compact set K ⊂ R n , we have lim ε → sup v ∈ K (cid:12)(cid:12)(cid:12) ( X z + ε v σ r − X z σ r ) /ε − A r v (cid:12)(cid:12)(cid:12) = 0 , a.s. Note that in the above theorem, both processes are observed at the same random time σ r ,the inverse local time for the process X z . In other words, we do not consider( X z + ε v σ z ε v r − X z σ z r ) /ε. The proof of the theorem will consist of several lemmas. We start by introducing somenotation.We will prove the theorem only for r = 1, and we will suppress r in the notation from nowon. It is clear that the same proof applies to any other value of r .It follows from Lemma 3.2 below that we can find a constant c ∗ and a sequence of stoppingtimes e T k such that e T k → ∞ , a.s., and sup z ∈ D L z e T k ≤ kc ∗ for all k . We fix some integer k ∗ ≥ σ ∗ = σ ∧ e T k ∗ . The dependence of σ ∗ on k ∗ and c ∗ will be suppressed in the notation.In much of the paper, we will consider “fixed” starting points z and y . We will write X t = X z t and Y t = X yt , so that X = z and Y = y . Later in this section, we will often take ε = | X − Y | . Let τ + δ = τ + ( δ ) = inf { t > | X t − Y t | ≥ δ } .We fix some (small) a , a >
0. We will impose some conditions on the values of a and a later on. Let S = U = inf { t ≥ X t ∈ ∂D } and for k ≥ S k = inf n t ≥ U k − : d ( X t , ∂D ) ∨ d ( Y t , ∂D ) ≤ a | X t − Y t | o ∧ σ ∗ , (3.1) U k = inf { t ≥ S k : | X t − X S k | ∨ | Y t − Y S k | ≥ a | X S k − Y S k |} ∧ σ ∗ . The filtration generated by the driving Brownian motion will be denoted F t . As usual,for a stopping time T , F T will denote the σ -field of events preceding T .Since D is bounded and ∂D is C , there exists δ > x ∈ D and d ( x, ∂D ) < δ then there is only one point y ∈ ∂D with | x − y | = d ( x, ∂D ). We will call this pointΠ x = Π( x ). For all other points, we let Π x = z ∗ , where z ∗ ∈ ∂D is a fixed reference point. tochastic flow of reflected Brownian motions 8 We define (random) linear operators, G k = exp (cid:16) ( L U k − L S k ) S (Π( X S k )) (cid:17) π Π( X Sk ) , (3.2) H k = exp (cid:16) ( L S k +1 − L S k ) S (Π( X S k )) (cid:17) π Π( X Sk ) . Recall the notation for excursions from Section 2.3. For ε ∗ >
0, let n e t ∗ , e t ∗ , . . . , e t ∗ m ∗ o = { e t ∈ E : | e t (0) − e t ( ζ − ) | ≥ ε ∗ , t < σ ∗ } . We label the excursions so that t ∗ k < t ∗ k +1 for all k and we let ℓ ∗ k = L t ∗ k for k = 1 , . . . , m ∗ .We also let t ∗ = inf { t ≥ X t ∈ ∂D } , ℓ ∗ = 0, ℓ ∗ m ∗ +1 = L σ ∗ , and ∆ ℓ ∗ k = ℓ ∗ k +1 − ℓ ∗ k . Let x ∗ k = e t ∗ k ( ζ − ) for k = 1 , . . . , m ∗ , and x ∗ = X t ∗ . Let γ ∗ ( s ) = x ∗ k for s ∈ [ ℓ ∗ k , ℓ ∗ k +1 ) and k = 0 , , . . . , m ∗ , and γ ∗ (1) = γ ∗ ( ℓ ∗ m ∗ ). Let I k = exp(∆ ℓ ∗ k S ( x ∗ k )) π x ∗ k . (3.3)Let ξ k = t ∗ k + ζ ( e t ∗ k ) for k = 1 , . . . , m ∗ , and ξ = 0.Let m ′ be the largest integer such that S m ′ ≤ σ ∗ . We let ℓ ′ k = L S k for k = 1 , . . . , m ′ . Wealso let t ′ = inf { t ≥ X t ∈ ∂D } , ℓ ′ = 0, ℓ ′ m ′ j +1 = L σ ∗ , and ∆ ℓ ′ k = ℓ ′ k +1 − ℓ ′ k . Note that wemay have ∆ ℓ ′ k = 0 for some k , with positive probability. Let x ′ k = Π( X S k ) for k = 1 , . . . , m ′ ,and x ′ = X t ′ . Let γ ′ ( s ) = x ′ k for s ∈ [ ℓ ′ k , ℓ ′ k +1 ) and k = 0 , , . . . , m ′ , and γ ′ (1) = γ ′ ( ℓ ′ m ′ ).Let λ : [0 , → [0 ,
1] be an increasing homeomorphism with the following properties. If t ∗ j = σ ℓ ∗ j ∈ ( U k , S k +1 ] for some j and k then we let λ ( ℓ ∗ j ) = ℓ ′ k +1 . For all other j , λ ( ℓ ∗ j ) = ℓ ∗ j .Let ℓ ′′ k = λ ( ℓ ∗ k ) for k = 1 , . . . , m ′′ := m ∗ . We also let t ′′ k = t ∗ k for k = 0 , , . . . , m ′′ , ℓ ′′ = 0, ℓ ′′ m ′′ j +1 = L σ ∗ , and ∆ ℓ ′′ k = ℓ ′′ k +1 − ℓ ′′ k . Let x ′′ k = x ∗ k for k = 0 , , . . . , m ′′ . Let γ ′′ ( s ) = x ′′ k for s ∈ [ ℓ ′′ k , ℓ ′′ k +1 ) and k = 0 , , . . . , m ′′ , and γ ′′ (1) = γ ′′ ( ℓ ′′ m ′′ ). Let J k = exp(∆ ℓ ′′ k S ( x ′′ k )) π x ′′ k . Note that ξ k = t ′′ k + ζ ( e t ′′ k ). Lemma 3.2.
There exists c and c , depending only on D , such that if for some integer m < ∞ and a sequence s < s < · · · < s m we have sup s k ≤ s,t ≤ s k +1 | B t − B s | ≤ c for k = 0 , , . . . , m − , then sup z ∈ D L zs m ≤ mc . Therefore, for every u < ∞ , we have sup z ∈ D L zu < ∞ , a.s.Proof. Let ν > r be as in Remark 2.3. We can suppose without loss of generalitythat 1 / (2 ν ) < r . Let r = 1 / (64 ν ). Then, by (2.8), for | x − y | ≤ r , x, y ∈ ∂D , we have | h n ( x ) , n ( y ) i − | ≤ νr < /
2, and, therefore, h n ( x ) , n ( y ) i ≥ /
2. Suppose that for some t and ω , sup ≤ s,t ≤ t | B t − B s | ≤ r /
64. Consider any z ∈ D and let t = inf { t ≥ X zt ∈ ∂D } ∧ t and y = X zt . If t = t then L zt = 0. tochastic flow of reflected Brownian motions 9 Suppose that t < t . Let t = inf { t ≥ t : | X zt − y | ≥ r }∧ t , t = sup { t ≤ t : X zt ∈ ∂D } and z = X zt . Then | z − y | ≤ / (64 ν ), so, by (2.10), |h z − y , n ( y ) i| ≤ ν/ (64 ν ) =1 / (64 ν ) = r / X zt − X zt = B t − B t for t ∈ [ t , t ], so sup t ≤ s,t ≤ t | X zt − X zs | ≤ r /
64. This impliesthat D X zt − X zt , n ( y ) E = D X zt − y , n ( y ) E (3.4)= D X zt − z , n ( y ) E + h z − y , n ( y ) i = D X zt − X zt , n ( y ) E + h z − y , n ( y ) i≤ r /
64 + r /
64 = r / . This implies that(1 / L zt − L zt ) ≤ (cid:28)Z t t n ( X zt ) dL zt , n ( y ) (cid:29) (3.5)= D X zt − X zt − ( B t − B t ) , n ( y ) E = D X zt − X zt , n ( y ) E − h ( B t − B t ) , n ( y ) i≤ r /
32 + r / < r / . Thus (cid:12)(cid:12)(cid:12) π y (cid:16) X zt − X zt (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) π y (cid:18) B t − B t + Z t t n ( X zt ) dL zt (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | B t − B t | + ( L zt − L zt ) ≤ r /
64 + r / < r / . This and (3.4) imply that | X zt − y | = | X zt − X zt | ≤ (( r / + ( r / ) / < r / . In view of the definition of t , we see that t = t . Hence, (3.5) shows that L zt = L zt − L zt ≤ r /
8. For a fixed ω , the above argument applies to all z ∈ D simultaneously, sosup z ∈ D L zt ≤ r / m < ∞ and a sequence 0 = s < s < · · · < s m , wehave sup s k ≤ s,t ≤ s k +1 | B t − B s | ≤ r /
64 for k = 0 , , . . . , m −
1. We can repeat the aboveargument on each interval [ s k , s k +1 ] to obtain sup z ∈ D L zs k +1 − L zs k ≤ r /
8, and, consequently,sup z ∈ D L zs m ≤ mr /
8. This proves the first assertion of the lemma.By continuity of Brownian motion, for any fixed u , with probability 1, one can finda (random) integer m < ∞ and a sequence 0 = s < s < · · · < s m = u such thatsup s k ≤ s,t ≤ s k +1 | B t − B s | ≤ r /
64 for k = 0 , , . . . , m −
1. The second assertion of the lemmafollows from this and the first part of the lemma. (cid:3)
Recall σ ∗ defined at the beginning of this section. tochastic flow of reflected Brownian motions 10 Lemma 3.3.
There exists c such that a.s., for all t ≤ σ ∗ and y, z ∈ D , we have | X yt − X zt | Let τ D = inf { t ≥ X t / ∈ D } and τ B ( x,r ) = inf { t ≥ X t / ∈ B ( x, r ) } .(i) There exists c such that if X = z ∈ D and d ( z , ∂D ) ≤ r then, P ( τ B ( z ,r ) ≤ τ D ) ≤ c d ( z , ∂D ) /r. (ii) Suppose d ( X , ∂D ) = b . Then E sup ≤ t ≤ τ D | X − X t | ≤ c b | log b | .Proof. (i) See Lemma 3.2 in [BCJ].(ii) By part (i), E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup ≤ t ≤ τ D X − X t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X b ≤ j ≤ diam( D ) j +1 P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup ≤ t ≤ τ D X − X t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∈ [2 j , j +1 ] ! ≤ X b ≤ j ≤ diam( D ) j +1 c b − j ≤ c b | log b | . (cid:3) Recall the notation from the beginning of this section. In particular, ε = | X − Y | . Lemma 3.5. For some c , E max ≤ k ≤ m ∗ sup ξ k ≤ t ≤ t ∗ k +1 | x ∗ k − X t | ≤ c ε / ∗ . (3.6) Proof. It follows from (3.19) in [BL] that, for any β < 1, some c , and all ε ∗ > E max ≤ k ≤ m ∗ sup ξ k ≤ t ≤ t ∗ k +1 ,X t ∈ ∂D | x ∗ k − X t | ≤ c ε β ∗ . (3.7) tochastic flow of reflected Brownian motions 11 The main difference between (3.6) and (3.7) is the presence of the condition X t ∈ ∂D in thesupremum. Let b E = { e ∈ E : | e (0) − e ( ζ − ) | < ε ∗ , sup ≤ t<ζ | e (0) − e ( t ) | ≥ ε ∗ } . Then max ≤ k ≤ m ∗ sup ξ k ≤ t ≤ t ∗ k +1 | x ∗ k − X t | ≤ max ≤ k ≤ m ∗ sup ξ k ≤ t ≤ t ∗ k +1 ,X t ∈ ∂D | x ∗ k − X t | (3.8)+ sup e ∈ b E sup ≤ t<ζ ( e ) | e (0) − e ( t ) | . Recall that n ≥ R n into which D is embedded. Standardestimates show that if T ∂D = inf { t ≥ X t ∈ ∂D } , x ∈ ∂D , y ∈ ∂ B ( x, r ) ∩ D , r > ρ , and X = y , then P ( X T ∂D ∈ B ( x, ρ ) ∩ ∂D ) ≤ c ( ρ/r ) n − . (3.9)We have for every x ∈ ∂D and b > c /b ≤ H x sup ≤ t<ζ ( e ) | e (0) − e ( t ) | ≥ b ! ≤ c /b. (3.10)The upper bound in the last estimate follows from (2.17) and Lemma 3.4 (i). The lowerbound can be proved in a similar way.We combine (3.9) and (3.10) using the strong Markov property of the measure H x appliedat the hitting time of B ( x, r ) to obtain, H x sup ≤ t<ζ ( e ) | e (0) − e ( t ) | ≥ ε / ∗ , | e (0) − e ( ζ − ) | < ε ∗ ! ≤ c ε − / ∗ c ( ε ∗ /ε / ∗ ) n − = c ε (2 / n − ∗ . By the exit system formula (2.16), P ∃ e ∈ b E : sup ≤ t<ζ | e (0) − e ( t ) | ≥ ε / ∗ ! ≤ c ε (2 / n − ∗ . So E sup e ∈ b E sup ≤ t<ζ ( e ) | e (0) − e ( t ) | ≤ ε / ∗ + diam( D ) P ∃ e ∈ b E : sup ≤ t<ζ | e (0) − e ( t ) | ≥ ε / ∗ ! ≤ ε / ∗ + diam( D ) c ε (2 / n − ∗ ≤ c ε / ∗ . The lemma follows by combining this estimate with (3.7) and (3.8). (cid:3) Lemma 3.6. There exists c such that if X ∈ ∂D then, E sup ≤ t ≤ ξ | X t − X ξ | ! ≤ c ε / ∗ . Proof. We havesup ≤ t ≤ ξ | X t − X ξ | ≤ max ≤ k ≤ m ∗ sup ξ k Recall that τ + δ = τ + ( δ ) = inf { t > | X t − Y t | ≥ δ } . Recall also that ε ∗ is the parameterused in the definition of ξ j and x ∗ j at the beginning of this section. Lemma 3.7. There exist c , . . . , c and ε , r , p > with the following properties. Let ε = ε ∧ r . Assume that X ∈ ∂D , | X − Y | = ε , d ( Y , ∂D ) = r and let T = inf { t ≥ | X t − X | ∨ | Y t − Y | ≥ c r } ,T = inf { t ≥ Y t ∈ ∂D } . ( T and T will be defined in the proof.)(i) If ε ≤ ε and r ≤ r then P ( S ≤ T ∧ T , L S − L ≤ c r ) ≥ p .(ii) If ε ≤ ε then E ( L S ∧ τ + ( ε ) − L ) ≤ c ( r + ε ) .(iii) If ε ≤ ε then E (sup ≤ t ≤ S ∧ τ + ( ε ) | X t − X | ) ≤ c | log r | ( r + ε ) .(iv) If ε ≤ ε and ε ∗ ≥ c ε then for any β < and all k , E X S k ≤ ξ j ≤ S k +1 ( L S k +1 − L ξ j ) | x ∗ j − Π( X S k +1 ) | | F S k ≤ c | X S k − Y S k | β . Remark 3.8. (i) Typically, we will be interested in small values of ε = | X − Y | . In viewof Lemma 3.3, | X t − Y t | ≤ c ε for all t ≤ σ ∗ . Hence, S ∧ τ + ( ε ) = S for ε much smallerthan ε . It follows that parts (ii) and (iii) of Lemma 3.7 can be applied with S in place of S ∧ τ + ( ε ), assuming small ε .(ii) The following remark applies to Lemma 3.7 and all other lemmas. Typically, theirproofs require that we assume that | X − Y | is bounded above. However, in many cases,the quantity that is being estimated is bounded above by a universal constant, for trivialreasons. Hence, by adjusting the constant appearing in the estimate, we can easily extendthe lemmas to all values of | X − Y | . tochastic flow of reflected Brownian motions 13 Proof of Lemma 3.7. (i) Recall ν defined in Remark 2.3. Assume that r < ε < / (200 ν ).Let c ∈ (0 , / 12) be a small constant whose value will be chosen later. Let T = inf { t ≥ h Y t − Y , n ( X ) i ≥ r } ,T = inf { t ≥ | π X ( Y t − Y ) | ≥ c r } ,A = { T ≤ T ∧ T } ,T = inf { t ≥ | π X ( X t − X ) | ≥ c r } . First we will assume that r ≤ ε / 2. We will show that T ≥ T ∧ T ∧ T if A holds.We will argue by contradiction. Assume that A holds and T < T ∧ T ∧ T . Then π X ( B t − B ) = π X ( Y t − Y ) for t ∈ [0 , T ] so | π X ( B t − B ) | ≤ c r for the same range of t ’s.We have π X ( X T − X ) = π X ( B T − B ) + Z T π X ( n ( X t )) dL t , so (cid:12)(cid:12)(cid:12)R T π X ( n ( X t )) dL t (cid:12)(cid:12)(cid:12) ≥ c r . By (2.12), we may assume that ε > r ≤ r < ε and x ∈ B ( X , c r ), we have | π X ( n ( x )) | ≤ νc r . This and the estimate (cid:12)(cid:12)(cid:12)R T π X ( n ( X t )) dL t (cid:12)(cid:12)(cid:12) ≥ c r imply that L T − L ≥ c r/ (4 νc r ) = 1 / (4 ν ). By (2.8), we maychoose ε so small that for r ≤ r < ε and x ∈ B ( X , c r ) ∩ ∂D , h n ( X ) , n ( x ) i ≥ / 2. Itfollows that * n ( X ) , Z T n ( X t ) dL t + ≥ / (8 ν ) . (3.13)By (2.9), we can assume that r and ε are so small that if for some y ∈ ∂D we have | π X ( y − X ) | ≤ c r then |h y − X , n ( X ) i| ≤ r ≤ ε ≤ ε . (3.14)Since d ( Y , ∂D ) = r , it is easy to see that if r > r ≤ r and t ≤ T ∧ T ∧ T , we have h Y t − Y , n ( X ) i ≥ − r , and, therefore, |h Y t − Y , n ( X ) i| ≤ r. (3.15)Note that h B t − B s , n ( X ) i = h Y t − Y s , n ( X ) i for s, t ∈ [0 , T ]. Since we have assumed that T < T ∧ T ∧ T , it follows that for s, t ∈ [0 , T ], |h B t − B s , n ( X ) i| = |h Y t − Y s , n ( X ) i| ≤ |h Y t − Y , n ( X ) i| + |h Y s − Y , n ( X ) i| ≤ r. (3.16)This, (2.14) and (3.13) imply that h X T − X , n ( X ) i ≥ −|h B T − B , n ( X ) i| + *Z T n ( X t ) dL t , n ( X ) + ≥ − r + 1 / (8 ν ) ≥ − ε + 1 / (8 ν ) ≥ ε . tochastic flow of reflected Brownian motions 14 Let T = sup { t ≤ T : X t ∈ ∂D } . The last estimate and (3.14) yield h B T − B T , n ( X ) i = h X T − X T , n ( X ) i = h X T − X , n ( X ) i + h X − X T , n ( X ) i≥ ε − ε = 22 ε , a contradiction with (3.16). This proves that T ≥ T ∧ T ∧ T if A holds. This and thedefinition of A imply that if A holds then T ≥ T .We will next show that if A holds then S ≤ T . Assume that A holds and let T =sup { t ≤ T : X t ∈ ∂D } . Note that neither X t nor Y t visit ∂D on the interval ( T , T ). Hence, X T − Y T = X T − Y T . If ε and r are sufficiently small then | π X ( X − Y ) | ≥ ε / r ≤ ε / d ( Y , ∂D ) = r . We have assumed that A holds so | π X ( Y T − Y ) | ≤ c r . Wehave proved that T ≥ T on A , so | π X ( X T − X ) | ≤ c r . Recall that c ≤ / 12 and r ≤ ε / 2. It follows that | X T − Y T | = | X T − Y T | ≥ | π X ( X T − Y T ) | (3.17) ≥ | π X ( X − Y ) | − | π X ( Y T − Y ) | − | π X ( X T − X ) |≥ ε / − c r − c r ≥ ε / . We have from the definition of T that | π X ( Y T − Y T ) | = | π X ( Y T − Y ) | + | π X ( Y − Y T ) | ≤ c r + c r = 2 c r. (3.18)The definition of T and (3.15) imply that for t ≤ T ∧ T ∧ T , | Y − Y t | ≤ r + c r < r. (3.19)Hence, | X − Y T | ≤ | X − Y | + | Y − Y T | ≤ ε + 3 r ≤ ε . (3.20)We have proved that T ≥ T on A , so | π X ( X T − X ) | ≤ c r ≤ ε . (3.21)Let x ∗ ∈ ∂D be the point with the minimal distance to Y T among points satisfying π X ( x ∗ ) = π X ( Y T ). We use the definition of x ∗ , (3.18), (3.20) and (2.13) to see that h Y T − x ∗ , n ( X ) i ≤ ν · c r · ε = 6 c νrε . (3.22)We use the fact that Y T − Y T = X T − X T and apply (2.13), (3.18) and (3.21), to obtain, h Y T − Y T , n ( X ) i = h X T − X T , n ( X ) i ≤ ν · c r · ε = 2 c νrε . We combine this estimate with (3.22) to see that d ( Y T , ∂D ) ≤ | Y T − x ∗ | = h Y T − x ∗ , n ( X ) i (3.23)= h Y T − Y T , n ( X ) i + h Y T − x ∗ , n ( X ) i ≤ c νrε + 6 c νrε = 8 c νrε . tochastic flow of reflected Brownian motions 15 This bound and (3.17) yield d ( Y T , ∂D ) | X T − Y T | ≤ c νrε ε / c rν ≤ c νε ≤ c ν | X T − Y T | . We make c > c ν ≤ a . Then d ( Y T , ∂D ) ≤ a | X T − Y T | .We obviously have d ( X T , ∂D ) ≤ a | X T − Y T | because X T ∈ ∂D . This shows that S ≤ T and completes the proof that if A holds then S ≤ T .Assume that A holds and suppose that D n ( X ) , R T n ( X t ) dL t E ≥ r . We will show thatthese assumptions lead to a contradiction. It follows from (3.15) that for s, t ≤ T ∧ T ∧ T , |h Y t − Y s , n ( X ) i| ≤ r. Since Y t − Y s = B t − B s for the same range of s and t , we obtain |h B t − B s , n ( X ) i| ≤ r. (3.24)This implies that h n ( X ) , X T − X i ≥ −|h n ( X ) , B T − B i| + * n ( X ) , Z T n ( X t ) dL t + ≥ − r + 20 r = 16 r. (3.25)Recall that T = sup { t ≤ T : X t ∈ ∂D } . In view of the definition of T and (3.14), h n ( X ) , X − X T i ≥ − r. (3.26)We have B T − B T = X T − X T so (3.25) and (3.26) give h n ( X ) , B T − B T i = h n ( X ) , X T − X T i = h n ( X ) , X T − X i + h n ( X ) , X − X T i ≥ r − r = 15 r. This contradicts (3.24) so we conclude that if A holds then * n ( X ) , Z T n ( X t ) dL t + ≤ r. (3.27)Note that h n ( X ) , n ( x ) i ≥ / x ∈ ∂D ∩ B ( X , c r ), assuming that ε > r ≤ r < ε . We have shown that if A holds then T ≥ T , so h n ( X ) , n ( X t ) i ≥ / t ∈ [0 , T ] such that X t ∈ ∂D . This and (3.27) imply that,(1 / L S − L ) ≤ (1 / L T − L ) ≤ * n ( X ) , Z T n ( X t ) dL t + ≤ r, and, therefore, L S − L ≤ r .By (3.24) and the fact that L T − L ≤ r , we have for t ≤ T , |h n ( X ) , X t − X i| ≤ |h n ( X ) , B t − B i| + (cid:28) n ( X ) , Z t n ( X t ) dL t (cid:29) ≤ r + 40 r = 44 r. This, the definition of T and the fact that T ≥ T on A imply that for t ≤ T , we have | X t − X | ≤ r . If we take c = 45 then this and (3.19) show that on A , T ≤ T and,therefore, S ≤ T ∧ T . tochastic flow of reflected Brownian motions 16 We proved that A ⊂ { S ≤ T ∧ T , L XS − L X ≤ r } . It is easy to see that P ( A ) > p for some p > c . This completes the proof of part (i) in the case r ≤ ε / 2, with c = 45 and c = 40.Next consider the case when r ≥ ε / 2. Let T = inf { t > | Y t − X | ≥ ε } ,T = inf { t > X t ∈ ∂D, d ( Y t , ∂D ) ≤ | X t − Y t | / } ,T = inf { t > L t − L ≥ ε } ,A = { T ≤ T } ,A = { T ≤ T ∧ T ∧ T } . We will show that A ⊂ A . Assume that A holds. Let T = inf { t ≥ | π X ( X t − X ) | ≥ ε } . We will show that T ≥ T . We will argue by contradiction. Assume that T < T . Wehave assumed that A holds, so T < T . Since T < T , we have π X ( B t − B ) = π X ( Y t − Y )and h n X , B t − B i = h n X , Y t − Y i for t ∈ [0 , T ], which implies in view of the definition of T that for s, t ∈ [0 , T ], | π X ( B t − B ) | = | π X ( Y t − Y ) | ≤ | π X ( Y t − X ) | + | π X ( Y − X ) | ≤ ε + ε = 3 ε , (3.28) |h n X , B t − B s i| = |h n X , Y t − Y s i| ≤ |h n X , Y t − X i| + |h n X , X − Y s i| ≤ ε + 2 ε = 4 ε . (3.29)We obtain from (3.28), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π X Z T n ( X t ) dL t !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | π X ( X T − X ) − π X ( B T − B ) | (3.30) ≥ | π X ( X T − X ) | − | π X ( B T − B ) | ≥ ε − ε = 2 ε . If ε > ε ≤ ε then by (2.12), | π X ( n ( x )) | ≤ νε for x ∈ ∂D ∩ B ( X , ε ). This and the estimate (cid:12)(cid:12)(cid:12)R T π X ( n ( X t )) dL t (cid:12)(cid:12)(cid:12) ≥ ε imply that L T − L ≥ ε / (10 νε ) = 1 / (5 ν ). By (2.8), we may choose ε so small that for ε ≤ ε and x ∈ B ( X , ε ) ∩ ∂D , h n ( X ) , n ( x ) i ≥ / 2. It follows that * n ( X ) , Z T n ( X t ) dL t + ≥ / (10 ν ) . Recall that ε < ε < / (200 ν ). We obtain from the last estimate and (3.29), h n X , X T − X i ≥ −|h n X , B T − B i| + * n X , Z T n ( X t ) dL t + ≥ − ε + 1 / (10 ν ) ≥ ε . Let T = sup { t ≤ T : X t ∈ ∂D } and note that, by (2.9), assuming ε is small, we have h n X , X − X t i ≥ − ε , (3.31) tochastic flow of reflected Brownian motions 17 for t ≤ T such that X t ∈ ∂D . Then h n X , B T − B T i = h n X , X T − X T i = h n X , X T − X i + h n X , X − X T i ≥ ε − ε = 15 ε . This contradicts (3.29) and, therefore, completes the proof that T ≥ T .Next we will prove that L T − L ≤ ε . Suppose otherwise, i.e., L T − L > ε . Wehave h n X , n ( x ) i ≥ / x ∈ ∂D ∩ B (0 , ε ), assuming ε > ε ≤ ε . Since T ≥ T , h n X , n ( X t ) i ≥ / t ≤ T such that X t ∈ ∂D , so, using (3.29), h n X , X T − X i ≥ −|h n X , B T − B i| + * n X , Z T n ( X t ) dL t + ≥ − ε + (1 / L T − L ) ≥ − ε + 10 ε = 6 ε . Recall that T = sup { t ≤ T : X t ∈ ∂D } and note that we can use (3.31) because T ≥ T ,so h n X , X − X T i ≥ − ε . Then h n X , B T − B T i = h n X , X T − X T i = h n X , X T − X i + h n X , X − X T i≥ ε − ε = 5 ε . This contradicts (3.29) because T ≤ T ≤ T . This proves that if A holds then L T − L ≤ ε ≤ r. (3.32)Recall the definition of T and the fact that T ≥ T to see that | π X ( X t − X ) | ≤ ε for t ≤ T , assuming that A holds. It follows from the definition of T that | Y t − Y | ≤ ε for t ≤ T . Recall that T = sup { t ≤ T : X t ∈ ∂D } . Note that X T − Y T = X T − Y T , Y T , X T ∈ ∂D , and T ≤ T . This and the bounds | π X ( X t − X ) | ≤ ε and | Y t − Y | ≤ ε for t ≤ T , easily imply that d ( Y T , ∂D ) ≤ | X T − Y T | / 2, assuming that ε is small. Hence, T ≤ T . This fact combined with (3.32) shows that if A occurs then T ≤ T ≤ T ∧ T .This completes the proof that A ⊂ A .It is easy to see that P ( A ) > p , for some p > 0. It follows that P ( A ) > p .We may now apply the strong Markov property at the stopping time T and repeat theargument given in the first part of the proof, which was devoted to the case r ≤ ε / 2. It isstraightforward to complete the proof of part (i), adjusting the values of c , c , ε , r and p ,if necessary.(ii) We will restart numbering of constants, i.e., we will use c , c , . . . , for constants unre-lated to those with the same index in the earlier part of the proof. tochastic flow of reflected Brownian motions 18 Let c , c , ε and r be as in part (i) of the lemma, ε = ε ∧ r , and ε ≤ ε . Recall that τ + ( ε ) = inf { t > | X t − Y t | ≥ ε } . Let T = 0, and for k ≥ T k = inf { t ≥ T k − : | X T k − − X t | ∨ | Y T k − − Y t | ≥ c d ( Y T k − , ∂D ) } ∧ τ + ( ε ) , (3.33) T k = inf { t ≥ T k − : L t − L T k − ≥ c d ( Y T k − , ∂D ) } ∧ τ + ( ε ) , (3.34) T k = inf { t ≥ T k − : Y t ∈ ∂D } ∧ τ + ( ε ) , (3.35) T k = T k ∧ T k ∧ T k , (3.36) T k = inf { t ≥ T k : X t ∈ ∂D } ∧ τ + ( ε ) . (3.37)We will estimate Ed ( Y T k , ∂D ). By Lemma 3.4 (i) and the definition of T k , on the event { T k < τ + ( ε ) } , P sup t ∈ [ T k ,T k ] | X t − X T k | ∈ [2 − j − , − j ] | F T k ≤ c d ( X T k , ∂D ) / − j ≤ c d ( Y T k − , ∂D ) / − j . (3.38)Write R = d ( Y T k − , ∂D ), assume that T k < τ + ( ε ), and let j be the largest integer suchthat sup t ∈ [ T k ,T k ] | X t − X T k |∨ ε ≤ − j . We will show that d ( Y T k , ∂D ) ≤ R + c ε − j , a.s. Notethat between times T k − and T k , the process Y t does not hit the boundary of D . Betweentimes T k and T k , the process X t does not hit ∂D . If Y t does not hit the boundary on thesame interval, it is elementary to see that d ( Y T k , ∂D ) ≤ R + c ε − j .Suppose that Y t ∗ ∈ ∂D for some t ∗ ∈ [ T k , T k ], and assume that t ∗ is the largest timewith this property. If t ∗ = T k then d ( Y T k , ∂D ) = 0. Otherwise we must have τ + ( ε ) > T k , X T k ∈ ∂D , and X T k − Y T k = X t ∗ − Y t ∗ . Since both Y t ∗ and X T k belong to ∂D , easygeometry shows that in this case d ( Y T k , ∂D ) ≤ c ε − j . This completes the proof that d ( Y T k , ∂D ) ≤ R + c ε − j , a.s.Let j be the smallest integer such that 2 − j ≥ diam( D ) and let j be the largest integersuch that 2 − j +1 ≥ R . The estimate d ( Y T k , ∂D ) ≤ R + c ε − j and (3.38) imply that on theevent { T k < τ + ( ε ) } , E ( d ( Y T k , ∂D ) | F T k ) ≤ X j ≤ j ≤ j ( R + c ε − j ) P ( sup t ∈ [ T k ,T k ] | X t − X T k | ∈ [2 − j − , − j ] | F T k ) ≤ R + X j ≤ j ≤ j c ε − j P ( sup t ∈ [ T k ,T k ] | X t − X T k | ∈ [2 − j − , − j ] | F T k ) ≤ R + X j ≤ j ≤ j c ε − j ( R/ − j ) ≤ R + c ε R | log R | = d ( Y T k − , ∂D )(1 + c ε | log d ( Y T k − , ∂D ) | ) . (3.39) tochastic flow of reflected Brownian motions 19 For R ≤ ε we have R (1+ c ε | log R | ) ≤ c ε , so R (1+ c ε | log R | ) ≤ R (1+4 c ε | log ε | )+ c ε . Thus, on the event { T k < τ + ( ε ) } , E ( d ( Y T k , ∂D ) | F T k ) ≤ (1 + c ε | log ε | ) d ( Y T k − , ∂D ) + c ε . (3.40)Let S ∗ = S ∧ τ + ( ε ). By the strong Markov property applied at T k − and part (i) of thelemma, on the event { S ∗ > T k − } , P ( T k − < S ∗ ≤ T k | F T k − ) ≥ P ( T k − < S ∗ ≤ T k | F T k − ) ≥ p . (3.41)By the strong Markov property and induction, P ( S ∗ > T k − ) ≤ c p k . (3.42)This, (3.40) and (3.41) imply, E (cid:16) d ( Y T k , ∂D ) { S ∗ >T k } { T k − <τ + ( ε ) } (cid:17) = E (cid:16) { S ∗ >T k } { T k − <τ + ( ε ) } E (cid:16) d ( Y T k , ∂D ) | F T k (cid:17)(cid:17) ≤ E (cid:16) { S ∗ >T k } { T k − <τ + ( ε ) } (cid:16) (1 + c ε | log ε | ) d ( Y T k − , ∂D ) + c ε (cid:17)(cid:17) = E (cid:16) { S ∗ >T k − } { S ∗ >T k } { T k − <τ + ( ε ) } (cid:16) (1 + c ε | log ε | ) d ( Y T k − , ∂D ) + c ε (cid:17)(cid:17) ≤ E (cid:18) { S ∗ >T k − } { T k − <τ + ( ε ) } (cid:16) (1 + c ε | log ε | ) d ( Y T k − , ∂D ) + c ε (cid:17) × E ( { S ∗ >T k } | F T k − ) (cid:19) ≤ E (cid:16) { S ∗ >T k − } { T k − <τ + ( ε ) } (cid:16) (1 + c ε | log ε | ) d ( Y T k − , ∂D ) + c ε (cid:17) (1 − p ) (cid:17) ≤ (1 + c ε | log ε | )(1 − p ) E (cid:16) d ( Y T k − , ∂D ) { S ∗ >T k − } { T k − <τ + ( ε ) } (cid:17) + c (1 − p ) ε P ( S ∗ > T k − ) ≤ (1 + c ε | log ε | )(1 − p ) E ( d ( Y T k − , ∂D ) { S ∗ >T k − } { T k − <τ + ( ε ) } )+ c (1 − p ) ε p k . tochastic flow of reflected Brownian motions 20 We assume without loss of generality that p > − p ) p − > 1. We obtainby induction, E ( d ( Y T k , ∂D ) { S ∗ >T k } { T k − <τ + ( ε ) } ) (3.43) ≤ (1 + c ε | log ε | ) k (1 − p ) k E ( d ( Y T , ∂D ) { S ∗ > } { T <τ + ( ε ) } )+ c (1 − p ) ε k − X m =0 (1 + c ε | log ε | ) m (1 − p ) m p k − m ≤ (1 + c ε | log ε | ) k (1 − p ) k r + c ε p k k − X m =0 (1 + c ε | log ε | ) m (1 − p ) m p − m ≤ (1 + c ε | log ε | ) k (1 − p ) k r + c ε p k (1 + c ε | log ε | ) k (1 − p ) k p − k = (1 + c ε | log ε | ) k (1 − p ) k r + c ε (1 + c ε | log ε | ) k (1 − p ) k ≤ c (1 + c ε | log ε | ) k (1 − p ) k ( r + ε ) . Note that, by (3.34) and (3.37), L T j +12 − L T j ≤ c d ( Y T j , ∂D ) ,L T j +15 − L T j +12 = 0 . Hence, L T j +15 − L T j ≤ c d ( Y T j , ∂D ) . (3.44)It follows from this and (3.43) that E ( L S ∧ τ + ( ε ) − L ) = E ( L S ∗ − L )= ∞ X k =0 E (cid:16) ( L S ∗ − L ) { S ∗ ∈ ( T k ,T k +15 ] } (cid:17) ≤ ∞ X k =0 E { S ∗ ∈ ( T k ,T k +15 ] } k X j =0 { T j <τ + ( ε ) } ( L T j +15 − L T j ) ≤ ∞ X k =0 E { S ∗ ∈ ( T k ,T k +15 ] } k X j =0 { T j − <τ + ( ε ) } c d ( Y T j , ∂D ) = E ∞ X k =0 k X j =0 { S ∗ ∈ ( T k ,T k +15 ] } { T j − <τ + ( ε ) } c d ( Y T j , ∂D ) = E ∞ X j =0 ∞ X k = j { S ∗ ∈ ( T k ,T k +15 ] } { T j − <τ + ( ε ) } c d ( Y T j , ∂D ) = c ∞ X j =0 E (cid:16) { S ∗ >T j } { T j − <τ + ( ε ) } d ( Y T j , ∂D ) (cid:17) ≤ ∞ X j =0 c (1 + c ε | log ε | ) j (1 − p ) j ( r + ε ) . If we assume that ε > c ( r + ε ). tochastic flow of reflected Brownian motions 21 (iii) We will restart numbering of constants, i.e., we will use c , c , . . . , for constantsunrelated to those with the same index in the earlier part of the proof.Recall that j is the largest integer such that 2 − j +1 ≥ d ( Y T k − , ∂D ). Let j be thelargest integer such that 2 − j +1 ≥ r . By (3.33) and (3.38) we have for j ≤ j , on the event { T k − < τ + ( ε ) } , P sup t ∈ [ T k − ,T k ] | X t − X T k | ∈ [2 − j − , − j ] | F T k − ≤ P sup t ∈ [ T k − ,T k ] | X t − X T k − | + sup t ∈ [ T k ,T k ] | X t − X T k | ∈ [2 − j − , − j ] | F T k − ≤ P c d ( Y T k − , ∂D ) + sup t ∈ [ T k ,T k ] | X t − X T k | ∈ [2 − j − , − j ] | F T k − ≤ c d ( Y T k − , ∂D ) / − j . We will also use the trivial estimate P sup t ∈ [ T k − ,T k ] | X t − X T k | ≤ r | F T k − ≤ . tochastic flow of reflected Brownian motions 22 We use the last two estimates, (3.42) and (3.43) to obtain E sup ≤ t ≤ S ∧ τ + ( ε ) | X t − X | ! = E sup ≤ t ≤ S ∗ | X t − X | ! = ∞ X k =0 E sup ≤ t ≤ S ∗ | X t − X | { S ∗ ∈ ( T k ,T k +15 ] } ! ≤ ∞ X k =0 E { S ∗ ∈ ( T k ,T k +15 ] } k X j =0 { T j <τ + ( ε ) } sup T j ≤ t ≤ T j +15 | X t − X | ≤ ∞ X k =0 E { S ∗ ∈ ( T k ,T k +15 ] } k X j =0 E { T j <τ + ( ε ) } sup T j ≤ t ≤ T j +15 | X t − X | | F T k − ≤ ∞ X k =0 E { S ∗ ∈ ( T k ,T k +15 ] } k X j =0 r + X j ≤ i ≤ j − i { T j − <τ + ( ε ) } c d ( Y T j − , ∂D ) / − i ≤ ∞ X k =0 E { S ∗ ∈ ( T k ,T k +15 ] } k X j =0 (cid:16) r + c | log r | { T j − <τ + ( ε ) } d ( Y T j − , ∂D ) (cid:17) = E ∞ X k =0 k X j =0 { S ∗ ∈ ( T k ,T k +15 ] } (cid:16) r + c | log r | { T j − <τ + ( ε ) } d ( Y T j − , ∂D ) (cid:17) = E ∞ X j =0 ∞ X k = j { S ∗ ∈ ( T k ,T k +15 ] } (cid:16) r + c | log r | { T j − <τ + ( ε ) } d ( Y T j − , ∂D ) (cid:17) = ∞ X j =0 E (cid:16) { S ∗ >T j } (cid:16) r + c | log r | { T j − <τ + ( ε ) } d ( Y T j − , ∂D ) (cid:17)(cid:17) = r ∞ X j =0 P ( S ∗ > T j ) + c | log r | ∞ X j =0 E (cid:16) { S ∗ >T j } { T j − <τ + ( ε ) } d ( Y T j − , ∂D ) (cid:17) ≤ r ∞ X j =0 c p k + c | log r | ∞ X j =0 (1 + c ε | log ε | ) j (1 − p ) j ( r + ε ) . If we assume that ε > c | log r | ( r + ε ).(iv) Once again, we will restart numbering of constants, i.e., we will use c , c , . . . , forconstants unrelated to those with the same index in the earlier part of the proof.Recall that j is the smallest integer such that 2 − j ≥ diam( D ). Let j be the smallest j with the property that 2 − j ≤ d ( Y T k , ∂D ). It follows from (3.38) that for any β < 1, on the tochastic flow of reflected Brownian motions 23 event { T k < τ + ( ε ) } , E sup T k ≤ t ≤ T k +15 | X T k − X t | | F T k ≤ E sup T k ≤ t ≤ T k +14 | X T k − X t | | F T k + E sup T k +14 ≤ t ≤ T k +15 | X T k +14 − X t | | F T k ≤ c d ( Y T k , ∂D ) + E sup T k +14 ≤ t ≤ T k +15 | X T k +14 − X t | | F T k ≤ c d ( Y T k , ∂D ) + j X j = j c − j d ( Y T k , ∂D ) / − j ≤ c d ( Y T k , ∂D )(1 + | log d ( Y T k , ∂D ) | ) ≤ c d ( Y T k , ∂D ) β ≤ c ε β . This and (3.43) imply that E d ( Y T k , ∂D ) { S ∗ >T k } { T k − <τ + ( ε ) } sup T k ≤ t ≤ T k +15 | X T k − X t | (3.45)= E d ( Y T k , ∂D ) { S ∗ >T k } { T k − <τ + ( ε ) } E sup T k ≤ t ≤ T k +15 | X T k − X t | | F T k ≤ c ε β E (cid:16) d ( Y T k , ∂D ) { S ∗ >T k } { T k − <τ + ( ε ) } (cid:17) ≤ c ε β (1 + c ε | log ε | ) k (1 − p ) k ( r + ε ) . It follows from the definition of S that | Π( X S ∗ ) − X S ∗ | ≤ c ε if S < σ ∗ ∧ τ + ( ε ). In thecase when S ∗ = σ ∗ ∧ τ + ( ε ), the distance between X and Y is increasing at this instance, soit is easy to see that the vector X S ∗ − Y S ∗ must also have a position such that | Π( X S ∗ ) − X S ∗ | ≤ c ε . (3.46)Recall that we assume that X ∈ ∂D , | X − Y | = ε , d ( Y , ∂D ) = r . Recall also that ε ∗ isthe parameter used in the definition of ξ j and x ∗ j at the beginning of this section. It followsfrom (3.33)-(3.37) that if ε ∗ ≥ c ε then at most one ξ i may belong to any given interval( T k − , T k ] and, moreover, if for some ξ i we have ξ i ∈ ( T k − , T k ] then ξ i = T k . This, (3.43), tochastic flow of reflected Brownian motions 24 (3.44), (3.45) and (3.46) imply that, E X ≤ ξ i ≤ S ∗ ( L S ∗ − L ξ i ) | x ∗ i − Π( X S ∗ ) | = ∞ X k =0 E X ≤ ξ i ≤ S ∗ ( L S ∗ − L ξ i ) | x ∗ i − Π( X S ∗ ) | { S ∗ ∈ ( T k ,T k +15 ] } ≤ ∞ X k =0 E { S ∗ ∈ ( T k ,T k +15 ] } k X j =0 { T j <τ + ( ε ) } { T j ≤ ξ i ≤ S ∗ } ( L T j +15 − L T j ) | x ∗ i − Π( X S ∗ ) | ≤ ∞ X k =0 E (cid:18) { S ∗ ∈ ( T k ,T k +15 ] } × k X j =0 { T j <τ + ( ε ) } { T j ≤ ξ i ≤ S ∗ } ( L T j +15 − L T j ) (cid:16) | X S ∗ − Π( X S ∗ ) | + | x ∗ i − X S ∗ | (cid:17) (cid:19) ≤ ∞ X k =0 E (cid:18) { S ∗ ∈ ( T k ,T k +15 ] } × (cid:18) k X j =0 ( j + 1) { T j <τ + ( ε ) } ( L T j +15 − L T j ) (cid:18) c ε + sup T j ≤ t ≤ T j +15 | X T j − X t | (cid:19)(cid:19)(cid:19) ≤ ∞ X k =0 E (cid:18) { S ∗ ∈ ( T k ,T k +15 ] } × (cid:18) k X j =0 ( j + 1) { T j <τ + ( ε ) } c d ( Y T j , ∂D ) (cid:18) c ε + sup T j ≤ t ≤ T j +15 | X T j − X t | (cid:19)(cid:19)(cid:19) = E ∞ X k =0 k X j =0 { S ∗ ∈ ( T k ,T k +15 ] } ( j + 1) { T j <τ + ( ε ) } c d ( Y T j , ∂D ) (cid:18) c ε + sup T j ≤ t ≤ T j +15 | X T j − X t | (cid:19) = E ∞ X j =0 ∞ X k = j { S ∗ ∈ ( T k ,T k +15 ] } ( j + 1) { T j <τ + ( ε ) } c d ( Y T j , ∂D ) (cid:18) c ε + sup T j ≤ t ≤ T j +15 | X T j − X t | (cid:19) = ∞ X j =0 E { S ∗ >T j } ( j + 1) { T j <τ + ( ε ) } c d ( Y T j , ∂D ) (cid:18) c ε + sup T j ≤ t ≤ T j +15 | X T j − X t | (cid:19) ≤ ∞ X j =0 c ( j + 1)(1 + c ε | log ε | ) j (1 − p ) j ( ε + ε β )( r + ε ) . If we assume that ε > c ε β ( r + ε ).Recall definitions of σ ∗ and S , and Lemma 3.3. There exists c such that if ε ≤ c ε then σ ∗ < τ + ( ε ). Hence, if ε ≤ c ε then E X ≤ ξ i ≤ S ( L S − L ξ i ) | x ∗ i − Π( X S ) | ≤ c ε β ( r + ε ) . (3.47) tochastic flow of reflected Brownian motions 25 Let b S k = inf { t ≥ S k : X t ∈ ∂D } ∧ σ ∗ . The following estimate can be proved just like(3.39), E (cid:16) d ( Y b S k , ∂D ) | F S k (cid:17) ≤ (1 + c ε | log ε | ) d ( Y S k , ∂D ) . We use this estimate, (3.47), the strong Markov property at b S k , and the definition of S k tosee that E X S k ≤ ξ j ≤ S k +1 ( L S k +1 − L ξ j ) | x ∗ j − Π( X S k +1 ) | | F S k = E Xb S k ≤ ξ j ≤ S k +1 ( L S k +1 − L ξ j ) | x ∗ j − Π( X S k +1 ) | | F S k = E E Xb S k ≤ ξ j ≤ S k +1 ( L S k +1 − L ξ j ) | x ∗ j − Π( X S k +1 ) | | F b S k | F S k ≤ E (cid:16) c | X b S k − Y b S k | β ( d ( Y b S k , ∂D ) + | X b S k − Y b S k | ) | F S k (cid:17) ≤ E (cid:16) c | X S k − Y S k | β ( d ( Y b S k , ∂D ) + | X S k − Y S k | ) | F S k (cid:17) ≤ c | X S k − Y S k | β ((1 + c ε | log ε | ) d ( Y S k , ∂D )) + | X S k − Y S k | ) ≤ c | X S k − Y S k | β (cid:16) (1 + c ε | log ε | ) | X S k − Y S k | (cid:17) + | X S k − Y S k | ) ≤ c | X S k − Y S k | β . (cid:3) Lemma 3.9. There exist c and a > such that for a , a < a , if | X − Y | = ε then a.s.,for every k ≥ , on the event U k < σ ∗ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* n (Π( X U k )) , Y U k − X U k | Y U k − X U k | +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε. Proof. It is elementary to see that one can choose c , a > ε > a < a , ε ≤ ε , x ∈ ∂D , y ∈ D , | x − y | ≤ ε , z ∈ ∂D , | x − z | ≤ a ε and | y − z | ≤ a ε , then * n ( z ) , y − x | y − x | + ≥ − c ε/ . (3.48)Moreover, if x, y ∈ D , w ∈ ∂D , | w − z | ≤ a ε and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* n ( z ) , y − x | y − x | +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε/ , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* n ( w ) , y − x | y − x | +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε. (3.49) tochastic flow of reflected Brownian motions 26 If | X − Y | = ε then | X t − Y t | ≤ c ε for all t ≤ σ ∗ , by Lemma 3.3. It follows easily from(3.1) that we can adjust the values of c and ε and choose a > | X − Y | = ε ≤ ε then on the event S k < σ ∗ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* n (Π( X S k )) , Y S k − X S k | Y S k − X S k | +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε/ . Let A = ( t ∈ [ S k , U k ] : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* n (Π( X S k )) , Y t − X t | Y t − X t | +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > c ε/ ) . We will show that A = ∅ . Suppose otherwise and let T = inf A . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* n (Π( X S k )) , Y T − X T | Y T − X T | +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = c ε/ . We must have either X T ∈ ∂D or Y T ∈ ∂D . It follows from (3.48) that either X T / ∈ ∂D or Y T / ∈ ∂D . Suppose without loss of generality that X T ∈ ∂D and Y T / ∈ ∂D . Then by(3.48), * n (Π( X S k )) , Y T − X T | Y T − X T | + = c ε/ . By the definition of T , for every δ > L t must increase on the interval [ T , T + δ ]. It iseasy to see that this implies that the function t → * n (Π( X S k )) , Y t − X t | Y t − X t | + is decreasing on the interval [ T , T + δ ], for some δ > 0. This contradicts the definition of T . Hence, for all t ∈ [ S k , U k ], (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* n (Π( X S k )) , Y t − X t | Y t − X t | +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε/ . In particular, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* n (Π( X S k )) , Y U k − X U k | Y U k − X U k | +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε/ . The lemma follows from the above estimate and (3.49). (cid:3) Lemma 3.10. There exists c such that if | X − Y | ≤ ε then for every k , E X U k ≤ σ ∗ ∧ τ + ( ε ) ( L S k +1 − L U k ) ≤ c ε | log ε | . Proof. We use the strong Markov property at the hitting time of ∂D by X and Lemma 3.7(ii) to see that E ( L S ∧ τ + ( ε ) − L U ) ≤ c ε. (3.50)We will estimate ( L S k +1 − L U k ) { U k <τ + ( ε ) } for k ≥ 1. Fix some k ≥ U k <τ + ( ε ). Note that d ( X U k , ∂D ) ≤ c | X U k − Y U k | . Let T = inf { t ≥ U k : X t ∈ ∂D } ∧ σ ∗ ∧ τ + ( ε ). tochastic flow of reflected Brownian motions 27 Let j be the greatest integer such that 2 − j is greater than the diameter of D and let j bethe least integer such that 2 − j ≤ | X U k − Y U k | . By Lemma 3.4, for j ≤ j ≤ j , P (cid:16) | X U k − X T | ∈ [2 − j , − j +1 ] | F U k (cid:17) ≤ c j | X U k − Y U k | . (3.51)Next we will estimate d ( Y T , ∂D ). Between times U k and T , the process X t does nothit ∂D . If Y t does not hit the boundary on the same interval, it is elementary to see fromLemma 3.9 that for j ≤ j ≤ j , d ( Y T , ∂D ) ≤ c | X U k − Y U k | + c | X U k − Y U k | − j ≤ c | X U k − Y U k | − j . Suppose that for some t ∗ ∈ [ U k , T ] we have Y t ∗ ∈ ∂D , and assume that t ∗ is the largest timewith this property. If t ∗ = T then d ( Y T , ∂D ) = 0. Otherwise we must have τ + ( ε ) > t ∗ , X T ∈ ∂D , and X T − Y T = X t ∗ − Y t ∗ . Since both Y t ∗ and X T belong to ∂D , easy geometryshows that in this case d ( Y T , ∂D ) ≤ c | X U k − Y U k | − j . We conclude that d ( Y T , ∂D ) ≤ c | X U k − Y U k | − j , a.s. By Lemma 3.7 (ii) and the strong Markov property applied at U k , E (cid:16) L S k +1 − L U k | U k < τ + ( ε ) , F T (cid:17) ≤ c ( | X U k − Y U k | − j + | X U k − Y U k | ) ≤ c | X U k − Y U k | − j . Hence, using (3.51), E (cid:16) L S k +1 − L U k | U k < τ + ( ε ) , F U k (cid:17) = E (cid:16) E (cid:16) L S k +1 − L U k | U k < τ + ( ε ) , F T (cid:17) F U k (cid:17) ≤ X j ≤ j ≤ j c | X U k − Y U k | j c | X U k − Y U k | − j ≤ c | X U k − Y U k | | log | X U k − Y U k || . It is elementary to check that E (cid:16) L U k − L S k | S k < τ + ( ε ) , F S k (cid:17) ≥ c | X S k − Y S k | , and the conditional distribution of L U k − L S k given { S k < τ + ( ε ) } is stochastically boundedby an exponential random variable with mean c | X S k − Y S k | . Note that | X U k − Y U k | ≤ c | X S k − Y S k | . Thus, E (cid:16) L S k +1 − L U k | U k < τ + ( ε ) , F U k (cid:17) ≤ c | X U k − Y U k | | log | X U k − Y U k || E (cid:16) L U k − L S k | S k < τ + ( ε ) , F S k (cid:17) ≤ c ε | log ε | E (cid:16) L U k − L S k | S k < τ + ( ε ) , F S k (cid:17) . It follows that N m := m X k =1 c ε | log ε | ( L U k − L S k ) { S k <τ + ( ε ) } − ( L S k +1 − L U k ) { U k <τ + ( ε ) } is a submartingale with respect to the filtration F ∗ m = F X,YS m +1 . If M = inf { m : m X k =1 ( L U k − L S k ) ≥ } tochastic flow of reflected Brownian motions 28 and M i = M ∧ i then E M i X k =1 (cid:16) c ε | log ε | ( L U k − L S k ) { S k <τ + ( ε ) } − ( L S k +1 − L U k ) { U k <τ + ( ε ) } (cid:17) ≥ , and E M i X k =1 ( L S k +1 − L U k ) { U k <τ + ( ε ) } ≤ E M i X k =1 c ε | log ε | ( L U k − L S k ) { S k <τ + ( ε ) } . We let i → ∞ and obtain by the monotone convergence E M X k =1 ( L S k +1 − L U k ) { U k <τ + ( ε ) } ≤ E M X k =1 c ε | log ε | ( L U k − L S k ) { S k <τ + ( ε ) } ≤ c ε | log ε | . Hence, E X k ≥ ,U k ≤ σ ∗ ∧ τ + ( ε ) ( L S k +1 − L U k ) ≤ E M X k =1 ( L S k +1 − L U k ) { U k <τ + ( ε ) } ≤ c ε | log ε | . This and (3.50) imply the lemma. (cid:3) Recall parameters a and a and operator G k defined in (3.2). Lemma 3.11. For any c there exist a , ε > such that if a , a ∈ (0 , a ) and | X − Y | = ε ≤ ε then a.s., the following holds for all k ≥ . Let Θ = Z U k S k n ( Y t ) dL yt − Z U k S k n (Π( Y S k )) dL yt ! (cid:16) | X S k − Y S k | · | L yU k − L yS k | (cid:17) − , with the convention that b/ . Then | Θ | ≤ c and (cid:12)(cid:12)(cid:12)(cid:12) G k ( Y S k − X S k ) − ( Y U k − X U k ) + ( n (Π( Y S k )) + Θ | X S k − Y S k | ) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17) + π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c | L U k − L S k | · | Y S k − X S k | . Proof. By (2.2), for any c , we can find ε > x, y ∈ ∂D with | x − y | ≤ ε , |S ( x ) π x ( x − y ) − ( n ( y ) − n ( x )) | ≤ ( c / | y − x | . (3.52)By Lemma 3.3, if we choose a sufficiently small ε > | Y t − X t | ≤ ε for all t ≤ σ ∗ .Estimate (3.52) and C -smoothness of ∂D can be used to show that for any c one canchoose small a , a > ε > k ≥ t ∈ [ S k , U k ] such that X t ∈ ∂D , |S (Π( X S k )) π Π( X Sk ) ( X S k − Y S k ) − ( n (Π( Y S k )) − n ( X t )) | ≤ c | Y S k − X S k | . (3.53) tochastic flow of reflected Brownian motions 29 We obtain from (2.14) and the triangle inequality, (cid:12)(cid:12)(cid:12)(cid:12) ( Y U k − X U k ) − ( Y S k − X S k ) − S (Π( X S k )) π Π( X Sk ) ( X S k − Y S k ) | L U k − L S k |− ( n (Π( Y S k )) + Θ | X S k − Y S k | ) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z U k S k n ( Y t ) dL yt − Z U k S k n ( X t ) dL t − S (Π( X S k )) π Π( X Sk ) ( X S k − Y S k ) | L U k − L S k |− ( n (Π( Y S k )) + Θ | X S k − Y S k | ) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U k S k n ( Y t ) dL yt − Z U k S k n (Π( Y S k )) dL yt − Θ | X S k − Y S k | ( L yU k − L yS k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | Θ | | X S k − Y S k | ( L U k − L S k )+ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U k S k ( n (Π( Y S k )) − n ( X t )) dL t − S (Π( X S k )) π Π( X Sk ) ( X S k − Y S k ) | L U k − L S k | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) Z U k S k n (Π( Y S k )) dL yt − Z U k S k n (Π( Y S k )) dL t − n (Π( Y S k )) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) . The expression on the last line is equal to zero for elementary reasons, so (cid:12)(cid:12)(cid:12)(cid:12) ( Y U k − X U k ) − ( Y S k − X S k ) − S (Π( X S k )) π Π( X Sk ) ( X S k − Y S k ) | L U k − L S k |− ( n (Π( Y S k )) + Θ | X S k − Y S k | ) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U k S k n ( Y t ) dL yt − Z U k S k n (Π( Y S k )) dL yt − Θ | X S k − Y S k | ( L yU k − L yS k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | Θ | | X S k − Y S k | ( L U k − L S k )+ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U k S k ( n (Π( Y S k )) − n ( X t )) dL t − S (Π( X S k )) π Π( X Sk ) ( X S k − Y S k ) | L U k − L S k | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The first term on the right hand side is equal to 0 by the definition of Θ. It is easy to seethat this claim holds even if the definition of Θ involves the division by 0. We have obtained (cid:12)(cid:12)(cid:12)(cid:12) ( Y U k − X U k ) − ( Y S k − X S k ) − S (Π( X S k )) π Π( X Sk ) ( X S k − Y S k ) | L U k − L S k | (3.54) − ( n (Π( Y S k )) + Θ | X S k − Y S k | ) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | Θ | | X S k − Y S k | ( L U k − L S k )+ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U k S k ( n (Π( Y S k )) − n ( X t )) dL t − S (Π( X S k )) π Π( X Sk ) ( X S k − Y S k ) | L U k − L S k | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . It follows from the definitions of S k , U k and Π x that for sufficiently small a and a , wehave for t ∈ [ S k , U k ], | Y t − Π( Y S k ) | ≤ a | X S k − Y S k | , tochastic flow of reflected Brownian motions 30 and a similar formula holds for X in place of Y on the left hand side. Hence, by (2.7), forsome c , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U k S k n ( Y t ) dL yt − Z U k S k n (Π( Y S k )) dL yt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z U k S k | n ( Y t ) − n (Π( Y S k )) | dL yt ≤ Z U k S k c | Y t − Π( Y S k ) | dL yt ≤ Z U k S k c · a | X S k − Y S k | dL yt ≤ a c | X S k − Y S k | · | L yU k − L yS k | . This shows that if we take a sufficiently small then | Θ | ≤ c .We use (3.53) to derive the following estimate, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U k S k ( n (Π( Y S k )) − n ( X t )) dL t − S (Π( X S k )) π Π( X Sk ) ( X S k − Y S k ) | L U k − L S k | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.55) ≤ c | X S k − Y S k | · | L U k − L S k | . We combine (3.54)-(3.55) to see that (cid:12)(cid:12)(cid:12)(cid:12) ( Y U k − X U k ) − ( Y S k − X S k ) − S (Π( X S k )) π Π( X Sk ) ( X S k − Y S k ) | L U k − L S k | (3.56) − ( n (Π( Y S k )) + Θ | X S k − Y S k | ) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( c / c ) | X S k − Y S k | · | L U k − L S k | . For any c , we can choose small ε so that (cid:12)(cid:12)(cid:12)(cid:12) π Π( X Sk ) ( Y S k − X S k ) + S (Π( X S k )) π Π( X Sk ) ( X S k − Y S k ) | L U k − L S k |− exp(( L U k − L S k ) S (Π( X S k ))) π Π( X Sk ) ( Y S k − X S k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c | X S k − Y S k | · | L U k − L S k | . This and (3.56) imply that (cid:12)(cid:12)(cid:12)(cid:12) Y U k − X U k − G k ( Y S k − X S k ) − ( n (Π( Y S k )) + Θ | X S k − Y S k | ) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17) + π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Y U k − X U k − exp(( L U k − L S k ) S (Π( X S k ))) π Π( X Sk ) ( Y S k − X S k ) − ( n (Π( Y S k )) + Θ | X S k − Y S k | ) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17) + π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( c / c ) | X S k − Y S k | · | L U k − L S k | . We obtain the lemma by choosing sufficiently small c . (cid:3) tochastic flow of reflected Brownian motions 31 Lemma 3.12. If a is sufficiently small then for some c , ε > and all ε < ε , if | X − Y | = ε then a.s., for all k ≥ , | ( L yU k − L yS k ) − ( L U k − L S k ) | ≤ c | Y S k − X S k | . Proof. Let w = n (Π( X S k )). It follows from the definition of U k that | Π( X S k ) − X t | ∨ | Π( X S k ) − Y t | ≤ c | Y S k − X S k | , for t ∈ [ S k , U k ]. This and (2.8) imply that for some c and t ∈ [ S k , U k ],1 − c | Y S k − X S k | ≤ h n ( X t ) , w i ≤ , for t such that X t ∈ ∂D, (3.57)1 − c | Y S k − X S k | ≤ h n ( Y t ) , w i ≤ , for t such that Y t ∈ ∂D. (3.58)We appeal to (2.13) to see that if a is sufficiently small and y ∈ ∂D and z ∈ D are suchthat max( | z − X S k | , | y − Y S k | ) ≤ a | X S k − Y S k | then for some c , | h y − z, w i | ≤ c | Y S k − X S k | , (3.59)and | h Y S k − X S k , w i | ≤ c | Y S k − X S k | . (3.60)Let I = { t ∈ [ S k , U k ] : h Y t − X t , w i ≥ c | Y S k − X S k | } . We claim that I = ∅ . Supposeotherwise and let t = inf I and t = sup { t ∈ [ S k , t ] : Y t ∈ ∂D } , with the convention thatsup ∅ = S k . By (3.57), (3.59) and (3.60), h Y t − X t , w i = h Y t − X t , w i + (cid:28)Z t t n ( Y s ) dL ys , w (cid:29) − (cid:28)Z t t n ( X s ) dL s , w (cid:29) ≤ h Y t − X t , w i + (cid:28)Z t t n ( Y s ) dL ys , w (cid:29) = h Y t − X t , w i ≤ c | Y S k − X S k | . This contradicts the definition of t , so we see that I = ∅ . Similarly, one can prove that { t ∈ [ S k , U k ] : h X t − Y t , w i ≥ c | Y S k − X S k | } = ∅ . Hence { t ∈ [ S k , U k ] : | h X t − Y t , w i | ≥ c | Y S k − X S k | } = ∅ . tochastic flow of reflected Brownian motions 32 This and (3.57)-(3.58) yield,(1 + c | Y S k − X S k | )( L yU k − L yS k ) − ( L U k − L S k ) ≤ *Z U k S k n ( Y s ) dL ys , w + − *Z U k S k n ( X s ) dL s , w + = h ( Y U k − Y S k ) − ( X U k − X S k ) , w i≤ c | Y S k − X S k | . By the definition of σ ∗ , L yU k − L yS k ≤ c , so the above estimate implies( L yU k − L yS k ) − ( L U k − L S k ) ≤ c | Y S k − X S k | + c | Y S k − X S k | ( L yU k − L yS k ) ≤ c | Y S k − X S k | . An analogous argument gives( L U k − L S k ) − ( L yU k − L yS k ) ≤ c | Y S k − X S k | . The lemma follows from the last two estimates. (cid:3) Lemma 3.13. For some c there exist a , ε > such that if a , a ∈ (0 , a ) , ε ≤ ε and | X − Y | = ε then for all k ≥ , E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:17)(cid:12)(cid:12)(cid:12) | F S k (cid:17) ≤ c ε | log ε | | Y S k − X S k | . Proof. The vector w k := π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) is parallel to n (Π( X S k )). It iseasy to check from the definition of S k that | w k | ≤ c | Y S k − X S k | .Let T = inf { t ≥ U k : X t ∈ ∂D } . It follows from Lemma 3.3 and definition of U k that d ( X U k , ∂D ) ≤ c ε . Let j be the smallest integer such that ε j is greater than the diameterof D . Lemma 3.4 (i) shows that for some c and all j = 1 , , . . . , j , P ( | X T − X U k | ≥ ε j | F U k ) ≤ c − j . By Lemma 3.7 (iii), the strong Markov property applied at T , and Chebyshev’s inequality, P ( | X T − X S k +1 | ≥ ε j | F T ) ≤ c ε | log ε | / ( ε j ) = c − j | log ε | . The fact that | X S k − X U k | ≤ c ε and the last two estimates show that P ( | X S k − X S k +1 | ≥ ε j | F S k ) ≤ c − j | log ε | . It is easy to see that | π Π( X Sk +1 ) w k | ≤ c ε j | w k | if | X S k − X S k +1 | ≤ ε j . It follows that E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:17)(cid:12)(cid:12)(cid:12) | F S k (cid:17) ≤ c ε | w k | + j X j =1 c ε j +1 | w k | P ( | X S k − X S k +1 | ∈ [ ε j , ε j +1 ] | F S k ) ≤ c εc | Y S k − X S k | + j X j =1 c ε j +1 c | Y S k − X S k | c − j | log ε |≤ c ε | log ε | | Y S k − X S k | . tochastic flow of reflected Brownian motions 33 (cid:3) Lemma 3.14. For some c there exist a , ε > such that if a , a ∈ (0 , a ) , ε ≤ ε and | X − Y | = ε then for all k ≥ , E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) | F U k (cid:17) ≤ c | Y U k − X U k | | log | Y U k − X U k || . Proof. Fix some k and let T = inf { t ≥ U k : X t ∈ ∂D or Y t ∈ ∂D } and ε = | X U k − Y U k | . We will assume from now on that X T ∈ ∂D . The rest of the argumentis similar if Y T ∈ ∂D .It follows from Lemma 3.3 and definition of U k that d ( X U k , ∂D ) ≤ c ε . Let j be thesmallest integer such that ε j is greater than the diameter of D . Lemma 3.4 shows thatfor some c and all j = 1 , , . . . , j , P ( | X T − X U k | ≥ ε j ) ≤ c − j . (3.61)By (2.9), we can choose c so small that for x ∈ ∂D ∩ B ( X T , c ε ), |h x − X T , n ( X T ) i| ≤ a ε / . (3.62)By the definition of σ ∗ , | Y t − X t | ≤ c ε for t ≤ σ ∗ . We make c smaller, if necessary, so that,in view of (2.11), |h y − x, n ( z ) i| ≤ a ε / , (3.63)assuming that x, y, z ∈ ∂D , | y − z | ≤ ( c + 5 c ) ε and | x − y | ≤ c ε .The following definitions contain a parameter c , the value of which will be chosen later.Let J = inf { j ≥ | X T − X U k | ≤ ε j } ,T = inf { t ≥ T : | B t − B T | ≥ c ε } ,T = inf { t ≥ T : h n ( X T ) , B t − B T i ≤ − c ε J } ,A = { T ≤ T } . Note that neither X nor Y touches the boundary of D between times U k and T , so Y T − X T = Y U k − X U k . Hence, by Lemma 3.9 and the strong Markov property applied at S k , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* n (Π( X U k )) , Y T − X T | Y T − X T | +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε . (3.64)The angle between n (Π( X U k )) and n ( X T ) is bounded by c ε J because ∂D is C . Thisand (3.64) imply that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* n ( X T ) , Y T − X T | Y T − X T | +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε J . (3.65) tochastic flow of reflected Brownian motions 34 Let k be such that c ε J ≤ / 10 if J ≤ k , and let F = { J ≤ k } . If F holds then (3.65)implies that, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π X T Y T − X T | Y T − X T | !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ / . (3.66) Case(i) . This case is devoted to an estimate of the random variable in the statement of thelemma assuming that A ∩ F holds. Since | Y T − X T | = ε , (3.65) implies that d ( Y T , ∂D ) ≤ c ε J . (3.67)Let c = 5 c and T = inf { t ≥ T : | X t − X T | ≥ c ε } ∧ T ∧ T ,T = sup { t ≤ T : X t ∈ ∂D } . We will show that T = T ∧ T , if ε (and, therefore, ε ) is sufficiently small. By (2.11), h x − y, n ( X T ) i ≤ c ε (3.68)for all x, y ∈ B ( X T , c ε ) such that x ∈ ∂D and y ∈ D . Since T ≤ T , we have h ( B T − B T ) , n ( X T ) i ≥ − c ε J . (3.69)This and (3.68) imply that *Z T T n ( X s ) dL s , n ( X T ) + = h ( X T − X T ) − ( B T − B T ) , n ( X T ) i ≤ c ε J . (3.70)For x ∈ ∂D ∩ B ( X T , c ε ) we have by (2.8), for small ε , h n ( x ) , n ( X T ) i ≥ − c ε ≥ / . (3.71)This and (3.70) show that L T − L T ≤ *Z T T n ( X s ) dL s , n ( X T ) + ≤ c ε J . (3.72)For x ∈ ∂D ∩ B ( X T , c ε ), (cid:12)(cid:12)(cid:12) π X T ( n ( x )) (cid:12)(cid:12)(cid:12) ≤ c ε . (3.73)It follows from this and (3.72) that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π X T Z T T n ( X s ) dL s !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε J ≤ c ε . (3.74)We can assume that ε is so small that for x ∈ ∂D ∩ B ( X T , c ε ), | x − X T | ≤ | π X T ( x − X T ) | . (3.75) tochastic flow of reflected Brownian motions 35 Since T ≤ T ∧ T , we can use (3.74) and (3.75) to obtain, | X T − X T | ≤ | X T − X T | + | X T − X T | ≤ | X T − X T | + 2 | π X T ( X T − X T ) | (3.76) ≤ | B T − B T | + 2 (cid:12)(cid:12)(cid:12) π X T ( B T − B T ) (cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π X T Z T T n ( X s ) dL s !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | B T − B T | + | B T − B T | + 2 (cid:12)(cid:12)(cid:12) π X T ( B T − B T ) (cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π X T Z T T n ( X s ) dL s !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε + 2 c ε . Recall that c = 5 c . Hence, the last estimate and the definition of T show that T = T ∧ T ,if ε is sufficiently small.Next we will estimate d ( X T , ∂D ). Let R = sup { t ≤ T : X t ∈ ∂D } . By the definition of T , h B T − B R , n ( X T ) i ≤ . This and the fact that X T − X R = B T − B R imply that, h X T − X R , n ( X T ) i ≤ . (3.77)Since X R ∈ ∂D ∩ B ( X T , c ε ), it follows from (3.62) and (3.77) that h X T − X T , n ( X T ) i = h X T − X R , n ( X T ) i + h X R − X T , n ( X T ) i ≤ a ε / . This and (3.62) imply that d ( X T , ∂D ) ≤ a ε / 800 = a ε / . (3.78)Our next goal is to estimate d ( Y T , ∂D ). Recall that | Y t − X t | ≤ c ε for t ≤ σ ∗ . Since T = T ∧ T , the definition of T implies that for t ∈ [ T , T ∧ T ], | Y t − X T | ≤ | Y t − X t | + | X t − X T | ≤ c ε + c ε = c ε . (3.79)Let c = 5 c and T = inf { t ≥ T : | Y t − Y T | ≥ c ε } ∧ T ∧ T . If Y t / ∈ ∂D for t ∈ [ T , T ] then L yT − L yT = 0. Suppose that Y t ∈ ∂D for some t ∈ [ T , T ]and let T = sup { t ≤ T : Y t ∈ ∂D } . We will show that T = T ∧ T , if ε (and, therefore, ε ) is sufficiently small. By (2.11), h x − y, n ( X T ) i ≤ c ε (3.80)for all x, y ∈ B ( X T , c ε ) such that x ∈ ∂D and y ∈ D . Since T ≤ T , we have h ( B T − B T ) , n ( X T ) i ≥ − c ε J . tochastic flow of reflected Brownian motions 36 Since T ≤ T ∧ T , we can use (3.80) and the last estimate to see that *Z T T n ( Y s ) dL ys , n ( X T ) + = h ( Y T − Y T ) − ( B T − B T ) , n ( X T ) i ≤ c ε J . (3.81)The above estimate is also valid in the case when Y t / ∈ ∂D for t ∈ [ T , T ] because in thiscase L yT − L yT = 0.For x ∈ ∂D ∩ B ( X T , c ε ) we have by (2.8), for small ε , h n ( x ) , n ( X T ) i ≥ − c ε ≥ / . This and (3.81) show that L yT − L yT ≤ *Z T T n ( Y s ) dL ys , n ( X T ) + ≤ c ε J . (3.82)For x ∈ ∂D ∩ B ( X T , c ε ), we have (cid:12)(cid:12)(cid:12) π X T ( n ( x )) (cid:12)(cid:12)(cid:12) ≤ c ε . It follows from this and (3.82)that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π X T Z T T n ( Y s ) dL ys !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε J ≤ c ε . (3.83)We can assume that ε is so small that for x ∈ ∂D ∩ B ( X T , c ε ), | x − X T | ≤ | π X T ( x − X T ) | . (3.84)Since T ≤ T ∧ T , (3.83) and (3.84) imply that | Y T − Y T | ≤ | Y T − Y T | + | Y T − Y T | ≤ | Y T − Y T | + 2 | π X T ( Y T − Y T ) | (3.85) ≤ | B T − B T | + 2 (cid:12)(cid:12)(cid:12) π X T ( B T − B T ) (cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π X T Z T T n ( Y s ) dL ys !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | B T − B T | + | B T − B T | + 2 (cid:12)(cid:12)(cid:12) π X T ( B T − B T ) (cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π X T Z T T n ( Y s ) dL ys !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε + 2 c ε . Recall that c = 5 c . The last estimate and the definition of T show that T = T ∧ T , if ε is sufficiently small.If ε is small then, by (3.79), for t ∈ [ T , T ∧ T ], | Π( Y t ) − X T | ≤ | Y t − X T | ≤ c ε . For x ∈ ∂D ∩ B ( X T , c ε ), by (2.9), |h x − X T , n ( X T ) i| ≤ c ε , (3.86)so, in particular, |h Π( Y T ) − X T , n ( X T ) i| ≤ c ε . tochastic flow of reflected Brownian motions 37 This and (3.67) imply that |h Y T − X T , n ( X T ) i| ≤ |h Π( Y T ) − X T , n ( X T ) i| + |h Π( Y T ) − Y T , n ( X T ) i| (3.87) ≤ c ε + c ε J ≤ c ε J . Recall that we assume that A holds so that T ≤ T . By (2.10), for x ∈ D ∩ B ( X T , c ε ), h x − X T , n ( X T ) i ≥ − c ε , so, in view of (3.79), h Y T − X T , n ( X T ) i ≥ − c ε . (3.88)We now choose the parameter c in the definition of T so that − c + c ≤ − c . Wewill show that given this choice of c , we must have Y t ∈ ∂D for t ∈ [ T , T ]. Suppose that Y t / ∈ ∂D for t ∈ [ T , T ]. Then Y t − Y T = B t − B T for the same range of t ’s. It follows from(3.87) and from the definition of T that h Y T − X T , n ( X T ) i = h Y T − Y T , n ( X T ) i + h Y T − X T , n ( X T ) i = h B T − B T , n ( X T ) i + h Y T − X T , n ( X T ) i≤ − c ε J + c ε J ≤ − c ε . This contradicts (3.88), so we conclude that Y must cross ∂D between times T and T .Hence, T is well defined. Since we are assuming that A holds, T ≤ T = T . Therefore, | Y T − Y T | ≤ | Y T − Y T | + | Y T − Y T | ≤ c ε = 10 c ε . (3.89)By (3.79), | Y T − X T | ≤ ( c + 5 c ) ε . This and (3.89) imply that the following can be derivedas a special case of (3.63), |h Y T − x, n ( X T ) i| ≤ a ε / , (3.90)for x ∈ ∂D ∩ B ( Y T , c ε ). By the definition of T , h B T − B T , n ( X T ) i ≤ . This and the fact that Y T − Y T = B T − B T imply that, h Y T − Y T , n ( X T ) i ≤ . We use this estimate and (3.90) to conclude that d ( Y T , ∂D ) ≤ a ε / . (3.91)Recall that we are assuming that F holds. It follows from (3.66) that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π X T Y T − X T | Y T − X T | !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ / , and, therefore, (cid:12)(cid:12)(cid:12) π X T ( Y T − X T ) (cid:12)(cid:12)(cid:12) ≥ ε / . tochastic flow of reflected Brownian motions 38 By (3.74) and (3.83) (cid:12)(cid:12)(cid:12) π X T ( Y T − X T ) (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) π X T ( Y T − X T ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π X T Z T T n ( X s ) dL s !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π X T Z T T n ( Y s ) dL ys !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) π X T ( Y T − X T ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π X T Z T T n ( X s ) dL s !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π X T Z T T n ( Y s ) dL ys !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ ε / − c ε − c ε . For small ε , this is bounded below by ε / 20. Hence, | Y T − X T | ≥ (cid:12)(cid:12)(cid:12) π X T ( Y T − X T ) (cid:12)(cid:12)(cid:12) ≥ ε / . This, (3.78) and (3.91) imply that S k +1 ≤ T , assuming A ∩ F holds.It follows from the definition of T and the fact that S k +1 ≤ T = T that | X S k +1 − X T | ≤ c ε . This implies that | Π( X S k +1 ) − X T | ≤ c ε , assuming that ε is sufficiently small.Let T = sup { t ∈ [ T , S k +1 ] : X t ∈ ∂D } . It is routine to check that (3.68)-(3.73) hold with X T replaced with Π( X S k +1 ), and T replaced with T (the values of the constants may have to be adjusted). Hence, we obtainas in (3.74) that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) Z S k +1 T n ( X s ) dL s !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) Z T T n ( X s ) dL s !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε J . (3.92)Similarly, an argument analogous to that in (3.80)-(3.83) yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) Z S k +1 T n ( Y s ) dL ys !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε J . This and (3.92) imply that (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) (3.93)= (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y T − X T ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) (3.94)= (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) Z S k +1 T n ( X s ) dL s − Z S k +1 T n ( Y s ) dL ys !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε J . We obtain from this and (3.61), E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) A ∩ F | F U k (cid:17) (3.95) ≤ j X j =1 c ε j − j ≤ c ε | log ε | = c ε | log ε | | Y U k − X U k | . tochastic flow of reflected Brownian motions 39 Case (ii) . We will now analyze the case when A does not occur. The rest of the proof isan outline only. Most steps are very similar to those in Case (i), so we omit details to savespace.Standard estimates show that P ( A c | F T ) ≤ c ε J . (3.96)Recall that we have assumed that X T ∈ ∂D . Let T = inf { t ≥ T : Y t ∈ ∂D } . For some c and c , we let K = inf { j ≥ t ∈ [ T ,T ] | Y t − Y T | ≤ ε j } ,T = inf { t ≥ T : | B t − B T | ≥ c ε } ,T = inf { t ≥ T : h n ( Y T ) , B t − B T i ≤ − c ε K } ,A = { T ≤ T } . Let T = sup { t ≤ T : X t ∈ ∂D } and note that X T − Y T = X T − Y T . Using the factthat X T ∈ ∂D and definitions of T , T and K , one can show that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* n ( Y T ) , Y T − X T | Y T − X T | +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* n ( Y T ) , Y T − X T | Y T − X T | +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε K . (3.97)This implies that d ( X T , ∂D ) ≤ c ε K . We can repeat the argument proving (3.94), withthe roles of X and Y interchanged and T replaced by T , to see that if A holds then S k +1 ≤ T and (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y T − X T ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ c ε K . (3.98)The angle between n ( Y T ) and n (Π( X S k +1 )) is less than c ε . We know from (3.67) that d ( Y T , ∂D ) ≤ c ε J . These facts and (3.97) imply that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* n (Π( X S k +1 )) , Z T T n ( X s ) dL s +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)D n (Π( X S k +1 )) , ( Y T − X T ) − ( Y T − X T ) E(cid:12)(cid:12)(cid:12) ≤ c ε J ∨ K . Let k be the largest integer such that if K ≤ k then for x ∈ ∂D ∩ B ( Y T , ε K ) we have h n ( x ) , n (Π( X S k +1 )) i ≥ / 2. Assume that F := { K ≤ k } holds. It follows that L T − L T ≤ *Z T T n ( X s ) dL s , n (Π( X S k +1 )) + ≤ c ε J ∨ K . We also have L T − L T ≤ c ε J by (3.72). Hence, L T − L T ≤ c ε J ∨ K .For x ∈ ∂D ∩ B ( Y T , ε K ), we have | π Π( X Sk +1 ) ( n ( x )) | ≤ c ε K , so (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) Z T T n ( X s ) dL s !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε ( J ∨ K )+ K . tochastic flow of reflected Brownian motions 40 By (3.82), L yT − L yT ≤ c ε J , so (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) Z T T n ( Y s ) dL ys !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) Z T T n ( Y s ) dL ys !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε J + K . Combining the last two estimates with (3.98), we obtain, (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y S k +1 − X S k +1 ) − ( Y T − X T ) (cid:17)(cid:12)(cid:12)(cid:12) (3.99)= (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y S k +1 − X S k +1 ) − ( Y T − X T ) (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (( Y T − X T ) − ( Y T − X T )) (cid:12)(cid:12)(cid:12) ≤ c ε K + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) Z T T n ( X s ) dL s !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) Z T T n ( Y s ) dL ys !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε ( J ∨ K )+ K . This implies that E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) A c ∩ A ∩ F | F U k (cid:17) (3.100)= j X j =1 j X k =1 E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) A c ∩ A ∩ F | J = j, K = k, F U k (cid:17) × P ( J = j, K = k | F U k ) . By (3.67) and an estimate similar to that in Lemma 3.4 (i), P ( K = k | F T ) ≤ c ε J ε − − k = c ε J − k . This, (3.61) an the strong Markov property applied at T yield, P ( J = j, K = k | F U k ) ≤ c − j ε j − k = c ε − k . (3.101)For K ≥ J we have 2 ( J ∨ K )+ K = 2 K so the the right hand side of (3.99) is bounded by c ε K . This and (3.101) imply that the corresponding contribution to the expectation in(3.100) is bounded by j X j =1 j X k = j c ε − k c ε k ≤ c ε | log ε | . (3.102)For K < J we have 2 ( J ∨ K )+ K = 2 J + K so the corresponding contribution to the expectationin (3.100) is bounded by j X j =1 j X k =1 c ε − k c ε j + k ≤ c ε | log ε | . Combining this with (3.102) yields E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) A c ∩ A ∩ F | F U k (cid:17) ≤ c ε | log ε | . (3.103)The probability that A does not occur, conditional on J and K , is bounded above by c ε K /ε = c ε K . If A c ∩ A c holds, we use the following crude estimate, (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ c ε . tochastic flow of reflected Brownian motions 41 Therefore, using (3.101), E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) A c ∩ A c | F U k (cid:17) (3.104) ≤ j X j =1 j X k =1 c ε − k c ε k c ε ≤ c ε | log ε | . It remains to address the cases when F or F fail. The probability of F c ∩ F c is boundedby c ε . Hence, E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) F c ∩ F c | F U k (cid:17) ≤ c ε c ε = c ε . (3.105)If F fails but F does not. we can repeat the analysis presented in Case (ii). Hence, (3.103)holds with A c ∩ A ∩ F replaced with F c ∩ A ∩ F . The lemma follows from these remarks, (3.95),(3.103), (3.104) and (3.105). (cid:3) Lemma 3.15. We have for some c , E m ′ X k =0 | Y S k − X S k | ≤ c . Proof. We will use modified versions of stopping times S k and U k by dropping σ ∗ from thedefinition (3.1). Let S ∗ = U ∗ = inf { t ≥ X t ∈ ∂D } and for k ≥ S ∗ k = inf n t ≥ U ∗ k − : d ( X t , ∂D ) ∨ d ( Y t , ∂D ) ≤ a | X t − Y t | o ,U ∗ k = inf n t ≥ S ∗ k : | X t − X S ∗ k | ∨ | Y t − Y S ∗ k | ≥ a | X S ∗ k − Y S ∗ k | o . Fix some k and let T = inf n t ≥ S ∗ k : D B t − B S ∗ k , n (Π( X S ∗ k )) E ≤ − ( a / | X S ∗ k − Y S ∗ k | o ,T = inf n t ≥ S ∗ k : D B t − B S ∗ k , n (Π( X S ∗ k )) E ≥ ( a / | X S ∗ k − Y S ∗ k | o ,T = inf (cid:26) t ≥ S ∗ k : (cid:12)(cid:12)(cid:12)(cid:12) π Π( X S ∗ k ) (cid:16) B t − B S ∗ k (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≥ ( a / | X S ∗ k − Y S ∗ k | (cid:27) ,A = { T ≤ T ≤ T } , F ∗ k = σ { B t , t ≤ S ∗ k } . Let ε = | X − Y | and recall that | X t − Y t | < c ε for t ≤ σ ∗ . By Brownian scaling and thestrong Markov property, P ( A | F ∗ k ) ≥ p on { S ∗ k ≤ σ ∗ } , for some p > ε or k . An argument similar to that in the proof of Lemma 3.7 (i) can be used to showthat if ε, a and a are small and A holds then T < U ∗ k and L T − L S ∗ k > ( a / | X S ∗ k − Y S ∗ k | .Then L U ∗ k − L S ∗ k > ( a / | X S ∗ k − Y S ∗ k | , so E ( L U ∗ k − L S ∗ k | F ∗ k ) > p ( a / | X S ∗ k − Y S ∗ k | . tochastic flow of reflected Brownian motions 42 We use this estimate to see that E m ′ X k =0 | Y S k − X S k | = E m ′ X k =0 | Y S ∗ k − X S ∗ k | (3.106)= E m ′ − X k =0 | Y S ∗ k − X S ∗ k | + | Y S ∗ m ′ − X S ∗ m ′ |≤ E m ′ − X k =0 c E (cid:16) L U ∗ k − L S ∗ k | F ∗ k (cid:17) + | Y S ∗ m ′ − X S ∗ m ′ |≤ c E m ′ − X k =0 (cid:16) L U ∗ k − L S ∗ k (cid:17) + | Y S ∗ m ′ − X S ∗ m ′ |≤ c E σ ∗ + | Y S ∗ m ′ − X S ∗ m ′ | . It is elementary to check that for all j , P ( L j +1 − L j > | σ { B t , t ≤ j } ) ≥ p > . Hence, σ ∗ ≤ σ is stochastically majorized by a geometric random variable with meandepending only on D , so E σ ∗ < c < ∞ . (3.107)We have | X S ∗ m ′ − Y S ∗ m ′ | < c ε because S ∗ m ′ ≤ σ ∗ . We combine this, (3.106) and (3.107) tocomplete the proof. (cid:3) Lemma 3.16. For some c there exists a > such that if a , a ∈ (0 , a ) and | X − Y | = ε then, E m ′ X k =0 | X S k − Y S k | (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17) ≤ c ε . Proof. We have by Lemmas 3.12 and 3.15, E m ′ X k =0 | X S k − Y S k | (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17) ≤ c ε E m ′ X k =0 | X S k − Y S k | ≤ c ε . (cid:3) Lemma 3.17. For some c there exists a > such that if a , a ∈ (0 , a ) and | X − Y | = ε then, E m ′ X k =0 (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) n (Π( Y S k )) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) ≤ c ε | log ε | . Proof. First, we will show that E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) ( n (Π( Y S k )) (cid:12)(cid:12)(cid:12) | F U k (cid:17) ≤ c | Y S k − X S k | | log | Y S k − X S k || . (3.108) tochastic flow of reflected Brownian motions 43 Recall the notation from the proof of Lemma 3.14, in particular, ε = | Y U k − X U k | , andnote that by Lemma 3.3, ε ≤ c | Y S k − X S k | . If A occurs then S k +1 ≤ T ≤ T . This anddefinitions of S k , U k , T , T and T imply that | Y S k − X S k +1 | ≤ | Y S k − X S k | + | X S k − X U k | + | X U k − X T | + | X T − X S k +1 |≤ c | Y S k − X S k | J . Therefore, (2.12) shows that (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) ( n (Π( Y S k )) (cid:12)(cid:12)(cid:12) ≤ c ε J . We calculate as in (3.95), E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) ( n (Π( Y S k )) (cid:12)(cid:12)(cid:12) A | F U k (cid:17) ≤ j X j =1 c ε j − j ≤ c ε | log ε | . (3.109)We obtain from (3.96), E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) ( n (Π( Y S k )) (cid:12)(cid:12)(cid:12) A c | F U k (cid:17) ≤ E (cid:16) A c | F U k (cid:17) ≤ j X j =1 c ε j − j ≤ c ε | log ε | . This and (3.109) prove (3.108). By (3.108) and Lemma 3.12, E (cid:16)(cid:12)(cid:12)(cid:12) n (Π( Y S k )) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17)(cid:12)(cid:12)(cid:12) | F U k (cid:17) ≤ c | Y S k − X S k | | log | Y S k − X S k || . We use this estimate and Lemma 3.15 to conclude that E m ′ X k =0 (cid:12)(cid:12)(cid:12) n (Π( Y S k )) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ E m ′ X k =0 c ε | log ε || Y S k − X S k | ≤ c ε | log ε | . (cid:3) Lemma 3.18. For some c there exist a , ε > such that if a , a ∈ (0 , a ) , ε ≤ ε and | X − Y | = ε then, E m ′ X k =0 (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ c ε | log ε | . Proof. Lemmas 3.13 and 3.15 imply that E m ′ X k =0 (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ E m ′ X k =0 E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:17)(cid:12)(cid:12)(cid:12) | F S k (cid:17) ≤ E m ′ X k =0 c ε | log ε | | Y S k − X S k | ≤ E m ′ X k =0 c ε | log ε | | Y S k − X S k | ≤ c ε | log ε | . (cid:3) tochastic flow of reflected Brownian motions 44 Lemma 3.19. For any c , ε > there exist a > , a random variable Λ and c such thatif ε ∈ (0 , ε ) , a , a < a and | X − Y | = ε then | Λ | ≤ c ε , a.s., and E (cid:12)(cid:12)(cid:12)(cid:12) | ( Y σ ∗ − X σ ∗ ) − G m ′ ◦ · · · ◦ G ( Y − X ) | − Λ (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε | log ε | . Proof. Note that S m ′ +1 = σ ∗ . We have G m ′ ◦ · · · ◦ G ( Y − X ) − ( Y σ ∗ − X σ ∗ ) (3.110)= m ′ X k =0 G m ′ ◦ · · · ◦ G k +1 (cid:16) G k ( Y S k − X S k ) − ( Y S k +1 − X S k +1 ) (cid:17) = m ′ X k =0 G m ′ ◦ · · · ◦ G k +1 ( G k ( Y S k − X S k ) − ( Y U k − X U k )) (3.111)+ m ′ X k =0 G m ′ ◦ · · · ◦ G k +1 (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17) . Recall Θ from Lemma 3.11. By (2.3), Lemma 3.3 and the triangle inequality, we have thefollowing estimate for the first sum in (3.111), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ′ X k =0 G m ′ ◦ · · · ◦ G k +1 ( G k ( Y S k − X S k ) − ( Y U k − X U k )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c m ′ X k =0 |G k +1 ( G k ( Y S k − X S k ) − ( Y U k − X U k )) |≤ c m ′ X k =0 (cid:12)(cid:12)(cid:12)(cid:12) G k +1 (cid:18) G k ( Y S k − X S k ) − ( Y U k − X U k )+ ( n (Π( Y S k )) + Θ | X S k − Y S k | ) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17) + π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + c m ′ X k =0 (cid:12)(cid:12)(cid:12) G k +1 (cid:16) n (Π( Y S k )) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) + c m ′ X k =0 (cid:12)(cid:12)(cid:12) G k +1 (cid:16) Θ | X S k − Y S k | (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) + c m ′ X k =0 (cid:12)(cid:12)(cid:12) G k +1 (cid:16) π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:17)(cid:12)(cid:12)(cid:12) . tochastic flow of reflected Brownian motions 45 We combine this with (3.110) to obtain |G m ′ ◦ · · · ◦ G ( Y − X ) − ( Y σ ∗ − X σ ∗ ) | (3.112) ≤ c m ′ X k =0 (cid:12)(cid:12)(cid:12)(cid:12) G k +1 (cid:18) G k ( Y S k − X S k ) − ( Y U k − X U k ) (3.113)+ ( n (Π( Y S k )) + Θ | X S k − Y S k | ) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17) (3.114)+ π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (3.115)+ c m ′ X k =0 (cid:12)(cid:12)(cid:12) G k +1 (cid:16) n (Π( Y S k )) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) (3.116)+ c m ′ X k =0 (cid:12)(cid:12)(cid:12) G k +1 (cid:16) Θ | X S k − Y S k | (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) (3.117)+ c m ′ X k =0 (cid:12)(cid:12)(cid:12) G k +1 (cid:16) π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:17)(cid:12)(cid:12)(cid:12) (3.118)+ m ′ X k =0 (cid:12)(cid:12)(cid:12) G m ′ ◦ · · · ◦ G k +1 (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) . (3.119)We need the following elementary fact about any non-negative real numbers b , b and b .Suppose that b ≤ b + b . Let Λ = max(0 , b − b ). Then | Λ | ≤ b . Moreover, | b − Λ | ≤ b .To see this, suppose that b ≥ b . Then Λ = b − b and | b − Λ | = | b − ( b − b ) | = b .If b < b then Λ = 0 and | b − Λ | = | b | < b . We apply these observations to b equal to(3.112), b equal to the sum of the terms (3.116)-(3.119), and b equal to (3.113)-(3.115). Tofinish the proof of the lemma, it will suffice to prove that b ≤ c ε, a.s. , (3.120)and E b ≤ c ε | log ε | . (3.121)Fix an arbitrarily small c > 0. By Lemma 3.3, | Y S k − X S k | ≤ c ε , for all k , a.s. ByLemma 3.11, if a and a are sufficiently small then with probability 1, b ≤ ( c /c ) m ′ X k =0 | L U k − L S k | · | Y S k − X S k | ≤ c ε m ′ X k =0 | L U k − L S k | . We have P m ′ k =0 | L U k − L S k | ≤ 1, so a.s., b ≤ c ε , that is, (3.120) holds true.We estimate (3.116) using (2.3) and Lemma 3.17, E m ′ X k =0 (cid:12)(cid:12)(cid:12) G k +1 (cid:16) n (Π( Y S k )) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) (3.122) ≤ c E m ′ X k =0 (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) n (Π( Y S k )) (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) ≤ c ε | log ε | . tochastic flow of reflected Brownian motions 46 Similarly, (2.3) and Lemma 3.18 yield the following estimate for (3.118), E m ′ X k =0 (cid:12)(cid:12)(cid:12) G k +1 (cid:16) π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:17)(cid:12)(cid:12)(cid:12) (3.123) ≤ c E m ′ X k =0 (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) π Π( X Sk ) ( Y S k − X S k ) − ( Y S k − X S k ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ c ε | log ε | . Recall from Lemma 3.11 that | Θ | ≤ c . By (2.3) and Lemmas 3.12 and 3.15, E m ′ X k =0 (cid:12)(cid:12)(cid:12) G k +1 (cid:16) Θ | X S k − Y S k | (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) (3.124) ≤ c E m ′ X k =0 (cid:12)(cid:12)(cid:12) | X S k − Y S k | (cid:16) ( L yU k − L yS k ) − ( L U k − L S k ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ c E m ′ X k =0 | X S k − Y S k | ≤ c ε E m ′ X k =0 | X S k − Y S k | ≤ c ε . By Lemma 3.14, E (cid:16)(cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) | F U k (cid:17) ≤ c | Y U k − X U k | | log | Y U k − X U k || . Hence, using (2.3) and Lemmas 3.3 and 3.15, E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ′ X k =0 G m ′ ◦ · · · ◦ G k +1 (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.125) ≤ c E m ′ X k =0 (cid:12)(cid:12)(cid:12) π Π( X Sk +1 ) (cid:16) ( Y U k − X U k ) − ( Y S k +1 − X S k +1 ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ c E m ′ X k =0 | Y U k − X U k | | log | Y U k − X U k || ≤ c ε | log ε | E m ′ X k =0 | Y U k − X U k | ≤ c ε | log ε | . The inequality in (3.121) follows from (3.122)-(3.125). This completes the proof of thelemma. (cid:3) Recall operator H k defined in (3.2). Lemma 3.20. For any c , ε > there exists a > such that if a , a < a and | X − Y | = ε then, E |G m ′ ◦ · · · ◦ G ( Y − X ) − H m ′ ◦ · · · ◦ H ( Y − X ) | ≤ c ε | log ε | . tochastic flow of reflected Brownian motions 47 Proof. We have G m ′ ◦ · · · ◦ G ( Y − X ) − H m ′ ◦ · · · ◦ H ( Y − X ) (3.126)= m ′ X k =0 G m ′ ◦ · · · ◦ G k +1 (cid:16) exp(( L U k − L S k ) S (Π( X S k ))) − exp(( L S k +1 − L S k ) S (Π( X S k ))) (cid:17) ◦ π Π( X Sk ) H k − ◦ · · · ◦ H ( Y − X ) . By (2.6), k exp(( L U k − L S k ) S (Π( X S k ))) − exp(( L S k +1 − L S k ) S (Π( X S k ))) k ≤ c | L U k − L S k +1 | . This, (2.3) and (3.126) imply that |G m ′ ◦ · · · ◦ G ( Y − X ) − H m ′ ◦ · · · ◦ H ( Y − X ) | ≤ c | ( Y − X ) | m ′ X k =0 | L U k − L S k +1 | . By Lemma 3.10, E P m ′ k =0 | L U k − L S k +1 | ≤ c ε | log ε | . Hence, E |G m ′ ◦ · · · ◦ G ( Y − X ) − H m ′ ◦ · · · ◦ H ( Y − X ) | ≤ c ε | log ε | . (cid:3) Recall notation from the beginning of this section. Lemma 3.21. We have for any β < and some c and c , assuming that | X − Y | = ε and ε ∗ ≥ c ε , E m ′ X k =0 X U k ≤ ξ j ≤ S k +1 ( L S k +1 − L ξ j ) | x ∗ j − Π( X S k +1 ) | ≤ c ε β . Proof. By Lemma 3.7 (iv), for every k , E X S k ≤ ξ j ≤ S k +1 ( L S k +1 − L ξ j ) | x ∗ j − Π( X S k +1 ) | | F S k ≤ c | X S k − Y S k | β . This and Lemma 3.15 imply that E m ′ X k =0 X U k ≤ ξ j ≤ S k +1 ( L S k +1 − L ξ j ) | x ∗ j − Π( X S k +1 ) | ≤ E m ′ X k =0 E X U k ≤ ξ j ≤ S k +1 ( L S k +1 − L ξ j ) | x ∗ j − Π( X S k +1 ) | | F S k ≤ E m ′ X k =0 c | X S k − Y S k | β ≤ E m ′ X k =0 c | X U k − Y U k | ε β ≤ c ε β . (cid:3) tochastic flow of reflected Brownian motions 48 For the notation used in the following lemma and its proof, see the beginning of thissection. Lemma 3.22. We have for any β < , some c and c , assuming that | X − Y | = ε and ε ∗ ≥ c ε , E |I m ∗ ◦ · · · ◦ I ( Y − X ) − J m ′′ ◦ · · · ◦ J ( Y − X ) | ≤ c ε β . Proof. We will follow closely the proof of Lemma 2.13 in [BL]. We will write S i = S ( x ′′ i ) = S ( x ∗ i ), π i = π x ′′ i = π x ∗ i . Recall that m ′′ = m ∗ . We have |J m ′′ ◦ · · · ◦ J ( Y − X ) − I m ∗ ◦ · · · ◦ I ( Y − X ) | = (cid:16) e ∆ ℓ ∗ m ∗ S m ∗ − e ( ℓ ∗ m ∗ +1 − ℓ ′′ m ∗ ) S m ∗ (cid:17) π m ∗ ◦ J m ′′ − ◦ · · · ◦ J ( Y − X )+ m ∗ X i =1 e ∆ ℓ ∗ m ∗ S m ∗ π m ∗ · · · e ∆ ℓ ∗ i +1 S i +1 π i +1 ◦ (cid:16) e ( ℓ ∗ i +1 − ℓ ′′ i ) S i π i e ∆ ℓ ′′ i − S i − − e ∆ ℓ ∗ i S i π i e ( ℓ ∗ i − ℓ ′′ i − ) S i − (cid:17) ◦ (3.127) π i − e ∆ ℓ ′′ i − S i − · · · e ∆ ℓ ′′ S π e ∆ ℓ ′′ S π ( Y − X )+ I m ∗ ◦ · · · ◦ I (cid:16) e ( ℓ ∗ − ℓ ′′ ) S − e ∆ ℓ ′′ S (cid:17) π ( Y − X ) . By virtue of (2.3) and (2.4), the last term is bounded by a constant multiple of | ℓ ∗ − ℓ ′′ | | Y − X | . Since ℓ ′′ ≥ ℓ ∗ , E | ℓ ∗ − ℓ ′′ | | Y − X | = ε E ( ℓ ′′ − ℓ ∗ ). By the strong Markovproperty applied at ξ and Lemma 3.7 (ii), E ( ℓ ′′ − ℓ ∗ ) ≤ c ε . Hence E (cid:16) I m ∗ ◦ · · · ◦ I (cid:16) e ( ℓ ∗ − ℓ ′′ ) S − e ∆ ℓ ′′ S (cid:17) π ( Y − X ) (cid:17) ≤ c E | ℓ ∗ − ℓ ′′ | | Y − X | ≤ c ε . (3.128)We have ℓ ′′ m ∗ +1 = ℓ ∗ m ∗ +1 = 1, so by (2.3) and (2.4), the first term on the right hand sideof (3.127) is bounded by a constant multiple of | ℓ ∗ m ∗ − ℓ ′′ m ∗ | | Y − X | . We have ℓ ′′ m ∗ ≥ ℓ ∗ m ∗ so E | ℓ ∗ m ∗ − ℓ ′′ m ∗ | | Y − X | ≤ ε E (1 − ℓ ∗ m ∗ ). The following estimate can be proved just like (3.10).We have for every x ∈ ∂D and b > c /b ≤ H x ( | e (0) − e ( ζ ) | ≥ b ) ≤ c /b. (3.129)This and the exit system formula (2.16) imply that 1 − ℓ ∗ is stochastically majorized by anexponential random variable with mean c ε , so E (1 − ℓ ∗ ) ≤ c ε . Hence E (cid:16)(cid:16) e ∆ ℓ ∗ m ∗ S m ∗ − e ( ℓ ∗ m ∗ +1 − ℓ ′′ m ∗ ) S m ∗ (cid:17) π m ∗ ◦ J m ′′ − ◦ · · · ◦ J ( Y − X ) (cid:17) (3.130) ≤ c E | ℓ ∗ m ∗ − ℓ ′′ m ∗ | | Y − X | ≤ c ε . The compositions before and after the parentheses in (3.127) in the summation are uni-formly bounded in operator norm by (2.3), so we need only estimate the sum m ∗ X i =0 (cid:13)(cid:13)(cid:13) e ( ℓ ∗ i +1 − ℓ ′′ i ) S i π i e ∆ ℓ ′′ i − S i − − e ∆ ℓ ∗ i S i π i e ( ℓ ∗ i − ℓ ′′ i − ) S i − (cid:13)(cid:13)(cid:13) . tochastic flow of reflected Brownian motions 49 Using the fact that π i commutes with S i , we can rewrite the i -th term in this sum as (cid:13)(cid:13)(cid:13) e ∆ ℓ ∗ i S i ◦ π i ◦ (cid:16) e ( ℓ ∗ i − ℓ ′′ i ) S i − e ( ℓ ∗ i − ℓ ′′ i ) S i − (cid:17) e ∆ ℓ ′′ i − S i − (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) e ∆ ℓ ∗ i S i (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) e ( ℓ ∗ i − ℓ ′′ i ) S i − e ( ℓ ∗ i − ℓ ′′ i ) S i − (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) e ∆ ℓ ′′ i − S i − (cid:13)(cid:13)(cid:13) . From (2.3) and (2.5), this last expression is bounded by c | ℓ ∗ i − ℓ ′′ i | (cid:12)(cid:12)(cid:12) x ′′ i − x ′′ i − (cid:12)(cid:12)(cid:12) . By Lemma3.21, for any β < E m ∗ X i =1 | ℓ ∗ i − ℓ ′′ i | (cid:12)(cid:12)(cid:12) x ′′ i − x ′′ i − (cid:12)(cid:12)(cid:12) ≤ c ε β . This combined with (3.128) and (3.130) yields the lemma. (cid:3) Once again, we ask the reader to consult the beginning of this section concerning notationused in the next lemma and its proof. Lemma 3.23. Suppose that ε ∗ = c ε , where c is as in Lemma 3.22. For some c , if weassume that | X − Y | = ε then, E |H m ′ ◦ · · · ◦ H ( Y − X ) − J m ′′ ◦ · · · ◦ J ( Y − X ) | ≤ c ε / | log ε | . Proof. Note that H k = exp(∆ ℓ ′ k ) S ( x ′ k )) π x ′ k . Let { ( ℓ k , x k ) } ≤ k ≤ m +1 be the sequence containing all the distinct elements of theunion of { ( ℓ ′ k , x ′ k ) } ≤ k ≤ m ′ +1 and { ( ℓ ′′ k , x ′′ k ) } ≤ k ≤ m ′′ +1 . We will explain how the sequence { ( ℓ k , x k ) } ≤ k ≤ m +1 is ordered but first we note that ℓ ′ k ’s need not be distinct, and neitherdo ℓ ′′ k ’s, and, moreover, some ℓ ′ k ’s may be equal to some ℓ ′′ k ’s. We order the sequence { ( ℓ k , x k ) } ≤ k ≤ m +1 in such a way that(i) ℓ k ≤ ℓ k +1 for all k .(ii) If ℓ k = ℓ ′ j , ℓ k = ℓ ′ j , ℓ ′ j = L S j , ℓ ′ j = L S j , and S j < S j then k < k .(iii) If ℓ k = ℓ ′′ j , ℓ k = ℓ ′′ j , ℓ ′′ j = λ ( ℓ ∗ j ), ℓ ′′ j = λ ( ℓ ∗ j ), and ℓ ∗ j < ℓ ∗ j then k < k .(iv) If ( ℓ k , x k ) = ( ℓ ′ j , x ′ j ), ( ℓ k , x k ) = ( ℓ ′′ j , x ′′ j ) and ℓ ′ j = ℓ ′′ j then k < k .It is easy to check that the above conditions define one and only one ordering of { ( ℓ k , x k ) } ≤ k ≤ m +1 .We introduce the following shorthand notations, ∆ i = ℓ i +1 − ℓ i , x i = γ ′ ( ℓ i ) , e x i = γ ′′ ( ℓ i ) , S i = S ( x i ) , e S i = S ( e x i ) ,π i = π x i , e π i = π e x i . tochastic flow of reflected Brownian motions 50 Observing that π e π = π and e π m +1 J m ′′ ◦ · · · ◦ J ( Y − X ) = J m ′′ ◦ · · · ◦ J ( Y − X ), wehave, H m ′ ◦ · · · ◦ H ( Y − X ) − J m ′′ ◦ · · · ◦ J ( Y − X )= m X i =0 e ∆ m S m π m · · · e ∆ i +1 S i +1 π i +1 (cid:18) e ∆ i S i π i − e π i +1 e ∆ i e S i (cid:19) e π i · · · e ∆ e S e π e ∆ e S e π ( Y − X ) . By (2.3), the compositions of operators before and after the parentheses in the summationabove are uniformly bounded in operator norm by a constant. Therefore, |H m ′ ◦ · · · ◦ H ( Y − X ) − J m ′′ ◦ · · · ◦ J ( Y − X ) | (3.131) ≤ c m X i =0 (cid:13)(cid:13)(cid:13)(cid:13) π i +1 ◦ (cid:18) e ∆ i S i ◦ π i − e π i +1 ◦ e ∆ i e S i (cid:19) ◦ e π i (cid:13)(cid:13)(cid:13)(cid:13) | Y − X | . Using the fact that S i and π i commute, as do e S i and e π i , we obtain, π i +1 ◦ (cid:18) e ∆ i S i ◦ π i − e π i +1 ◦ e ∆ i e S i (cid:19) ◦ e π i (3.132)= π i +1 ◦ π i ◦ (cid:18) e ∆ i S i − e ∆ i e S i (cid:19) ◦ e π i + π i +1 ◦ ( π i − e π i +1 ) ◦ e π i ◦ e ∆ i e S i . We will deal with each of these terms separately.For the first term, we have by (2.5), (cid:13)(cid:13)(cid:13)(cid:13) π i +1 ◦ π i ◦ (cid:18) e ∆ i S i − e ∆ i e S i (cid:19) ◦ e π i (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) e ∆ i S i − e ∆ i e S i (cid:13)(cid:13)(cid:13)(cid:13) ≤ c ∆ i | x i − e x i | . (3.133)For the second term on the right hand side of (3.132), Lemma 2.2 and (2.3) allow us toconclude that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) π i +1 ◦ ( π i − e π i +1 ) ◦ e π i ◦ e ∆ i e S i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ c ( | x i +1 − x i | | x i − e x i | + | x i +1 − e x i +1 | | e x i +1 − e x i | ) (cid:13)(cid:13)(cid:13)(cid:13) e ∆ i e S i (cid:13)(cid:13)(cid:13)(cid:13) ≤ c ( | x i +1 − x i | | x i − e x i | + | x i +1 − e x i +1 | | e x i +1 − e x i | ) . (3.134)We will now analyze (3.133). Suppose that ∆ i > x i = e x i . Let j and k be definedby x i = γ ′ ( ℓ ′ j ) and e x i = γ ′′ ( ℓ ′′ k ).Suppose that ℓ i = ℓ ′ j = ℓ ′′ k +1 . Then, by our ordering of ℓ r ’s, ℓ i +1 = ℓ ′′ k +1 = ℓ i , so ∆ i = 0.For the same reason, we have ∆ i = 0 if any of the following conditions holds: ℓ ′′ k = ℓ i = ℓ ′ j or ℓ i = ℓ ′′ k = ℓ ′ j +1 . For this reason we consider only sharp versions of the correspondinginequalities in (3.135)-(3.138) below.We have assumed that x i = e x i so one of the following four events holds, F i = { ℓ ′′ k < ℓ i = ℓ ′ j < ℓ ′′ k +1 , ξ k < S j ≤ t ′′ k +1 } , (3.135) F i = { ℓ ′′ k < ℓ i = ℓ ′ j < ℓ ′′ k +1 , t ′′ k +1 < S j ≤ ξ k +1 } , (3.136) F i = { ℓ ′ j < ℓ i = ℓ ′′ k < ℓ ′ j +1 , S j < ξ k ≤ U j ≤ S j +1 } , (3.137) F i = { ℓ ′ j < ℓ i = ℓ ′′ k < ℓ ′ j +1 , S j < U j ≤ ξ k ≤ S j +1 } . (3.138) tochastic flow of reflected Brownian motions 51 If F i holds then, { ξ k ≤ S j ≤ t ′′ k +1 } ∩ {| x i − e x i | > a } ⊂ [ ≤ r ≤ m sup ξ r Suppose that | Y − X | = ε and ε ∗ = c ε , where c is as in Lemma3.22. Consider an arbitrarily small c > ε > 0, we have a.s., | Λ | < c ε. (3.160)By the triangle inequality, | ( Y σ − X σ ) − I m ∗ ◦ · · · ◦ I ( Y − X ) | (3.161) ≤ | Λ | + (cid:12)(cid:12)(cid:12)(cid:12) | ( Y σ − X σ ) − G m ′ ◦ · · · ◦ G ( Y − X ) | − Λ (cid:12)(cid:12)(cid:12)(cid:12) + |G m ′ ◦ · · · ◦ G ( Y − X ) − H m ′ ◦ · · · ◦ H ( Y − X ) | + |H m ′ ◦ · · · ◦ H ( Y − X ) − J m ′′ ◦ · · · ◦ J ( Y − X ) | + |J m ′′ ◦ · · · ◦ J ( Y − X ) − I m ∗ ◦ · · · ◦ I ( Y − X ) | := | Λ | + Ξ . By Lemma 3.19, E (cid:12)(cid:12)(cid:12)(cid:12) | ( Y σ ∗ − X σ ∗ ) − G m ′ ◦ · · · ◦ G ( Y − X ) | − Λ (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε | log ε | . (3.162)By Lemma 3.20, E |G m ′ ◦ · · · ◦ G ( Y − X ) − H m ′ ◦ · · · ◦ H ( Y − X ) | ≤ c ε | log ε | . (3.163)Lemma 3.23 implies that E |H m ′ ◦ · · · ◦ H ( Y − X ) − J m ′′ ◦ · · · ◦ J ( Y − X ) | ≤ c ε / | log ε | . (3.164)Lemma 3.22 yields for any β < E |J m ′′ ◦ · · · ◦ J ( Y − X ) − I m ∗ ◦ · · · ◦ I ( Y − X ) | ≤ c ε β . (3.165)Combining (3.162)-(3.165), and using the definition of Ξ in (3.161), we see that E Ξ ≤ c ε / | log ε | . (3.166)Fix some β ∈ (1 , / 3) and β ∈ (0 , / − β ). By (3.166) and Chebyshev’s inequality, P (Ξ > c ε β ) ≤ c ε β . (3.167)Fix an arbitrary b > v ∈ R n with | v | = 1. We apply the last estimate to a sequenceof processes Y = X z + ε v with ε = b − k , k ≥ k , for some fixed large k . We obtain P (Ξ > c b − kβ ) ≤ c b − kβ , k ≥ k . tochastic flow of reflected Brownian motions 55 Since P k ≥ k c b − kβ < ∞ , the Borel-Cantelli Lemma shows that only a finite number ofevents { Ξ > c b − kβ } occur. This is the same as saying that only a finite number of events { Ξ /b − k > c b − k ( β − } occur. We combine this fact with (3.160) and (3.161) to see that forany c > 0, a.s., lim sup k →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + b − k v σ ∗ − X σ ∗ b − k − I m ∗ ◦ · · · ◦ I ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c . Since c is arbitrarily small, we have in fact, a.s.,lim k →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + b − k v σ ∗ − X σ ∗ b − k − I m ∗ ◦ · · · ◦ I ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (3.168)It is easy to see that the last formula holds for all v ∈ R n , not only those with | v | = 1.Consider an arbitrary compact set K ⊂ R n . Let c be the same constant as c in thestatement of Lemma 3.3. It follows easily from (2.3) that kI m ∗ ◦ · · · ◦ I k ≤ c , a.s. Fix any c > w , . . . , w j ∈ R n such that for every v ∈ K there exists j = j ( v ) such that | v − w j | < c / (2( c + c )). Note that | ( z + b − k v ) − ( z + b − k w j ( v ) ) | < b − k c / (2 c ) and, inview of (3.168), lim k →∞ sup ≤ j ≤ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + b − k w j σ ∗ − X σ ∗ b − k − I m ∗ ◦ · · · ◦ I ( w j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (3.169)By Lemma 3.3, for v ∈ K and j = j ( v ), a.s., (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + b − k w j σ ∗ − X σ ∗ b − k − X z + b − k v σ ∗ − X σ ∗ b − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c | ( z + b − k v ) − ( z + b − k w j ) | /b − k ≤ c / . (3.170)Since | v − w j | < c / (2 c ), |I m ∗ ◦ · · · ◦ I ( w j ( v ) ) − I m ∗ ◦ · · · ◦ I ( v ) | ≤ c / . (3.171)Combining (3.169)-(3.171) yields a.s.,lim k →∞ sup v ∈ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + b − k v σ ∗ − X σ ∗ b − k − I m ∗ ◦ · · · ◦ I ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c . Since c > k →∞ sup v ∈ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + b − k v σ ∗ − X σ ∗ b − k − I m ∗ ◦ · · · ◦ I ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (3.172)Let c = sup {| v | ∈ K } . For ε ∈ [ b − k , b − k +1 ), we have, | ( z + b − k v ) − ( z + ε v ) | /ε ≤ c (1 − /b ) . tochastic flow of reflected Brownian motions 56 Hence, by Lemma 3.3, a.s., (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + b − k v σ ∗ − X σ ∗ b − k − X z + ε v σ ∗ − X σ ∗ ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + b − k v σ ∗ − X σ ∗ b − k − X z + b − k v σ ∗ − X σ ∗ ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + b − k v σ ∗ − X σ ∗ ε − X z + ε v σ ∗ − X σ ∗ ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 − /b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + b − k v σ ∗ − X σ ∗ b − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + c | ( z + ε v ) − ( z + b − k v ) | /ε ≤ (1 − /b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + b − k v σ ∗ − X σ ∗ b − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + c c (1 − /b ) . Let ε ∗ = c b − k , where k is defined by ε ∈ [ b − k , b − k +1 ). The last formula and (3.172) yield,lim ε → sup v ∈ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + ε v σ ∗ − X σ ∗ ε − I m ∗ ◦ · · · ◦ I ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 − /b ) lim sup k →∞ sup v ∈ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + b − k v σ ∗ − X σ ∗ b − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + c c (1 − /b ) . Let ε ∗ = c ε . We can take b > ε → sup v ∈ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + ε v σ ∗ − X σ ∗ ε − I m ∗ ◦ · · · ◦ I ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . Recall the definition of σ ∗ from the beginning of this section. We let k ∗ → ∞ to see that,a.s., lim ε → sup v ∈ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X z + ε v σ − X σ ε − I m ∗ ◦ · · · ◦ I ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . We combine this with Theorem 2.5 to complete the proof of the theorem. (cid:3) References [A] H. 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