Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 2014, Vol. 50, No. 4, 1323–1346 DOI: 10.1214/13-AIHP541 © Association des Publications de l’Institut Henri Poincaré, 2014
www.imstat.org/aihp
Three examples of Brownian flows on R
Yves Le Jana and Olivier Raimondb
aLaboratoire Modélisation stochastique et statistique, Université Paris-Sud, Bâtiment 425, 91405 Orsay Cedex, France. E-mail: [email protected] bLaboratoire Modal’X, Université Paris Ouest Nanterre La Défense, Bâtiment G, 200 avenue de la République 92000 Nanterre, France. E-mail: [email protected] Received 21 November 2011; revised 14 January 2013; accepted 22 January 2013
Abstract. We show that the only flow solving the stochastic differential equation (SDE) on R = + + − dXt 1{Xt >0}W (dt) 1{Xt <0} dW (dt), + − ± ± where W and W are two independent white noises, is a coalescing flow we will denote by ϕ .Theflowϕ is a Wiener solution + = [ | +] of the SDE. Moreover, K E δϕ± W is the unique solution (it is also a Wiener solution) of the SDE t t + = + + + 1 Ks,tf(x) f(x) Ks,u 1R+ f (x)W (du) Ks,uf (x) du s 2 s + for s Résumé. Nous montrons que le seul flot solution de l’équation différentielle stochastique (EDS) sur R = + + − dXt 1{Xt >0}W (dt) 1{Xt <0} dW (dt), + − ± ± où W et W sont deux bruits blancs indépendants, est un flot coalescent que nous noterons ϕ .Leflotϕ est une solution + = [ | +] Wiener de l’équation. De plus, K E δϕ± W est l’unique solution (c’est aussi une solution Wiener) de l’EDS t t + = + + + 1 Ks,tf(x) f(x) Ks,u 1R+ f (x)W (du) Ks,uf (x) du s 2 s + pour tout s MSC: Primary 60H25; secondary 60J60 Keywords: Stochastic flows; Coalescing flow; Arratia flow or Brownian web; Brownian motion with oblique reflection on a wedge 1. Introduction Our purpose in this paper is to study two very simple one dimensional SDE’s which can be completely solved, although they do not satisfy the usual criteria. This study is done in the framework of stochastic flows exposed in 1324 Y. Le Jan and O. Raimond [11–13] and [20](seealso[16] and [17] in an SPDE setting). There is still a lot to do to understand the nature of these flows, even if one consider only Brownian flows, i.e. flows whose one-point motion is a Brownian motion. As in our previous study of Tanaka’s equation (see [14] and also [7,8]), it is focused on the case where the singularity is located at an interface between two half lines (see also [4] and references therein where a flow related to the skew Brownian motion is studied and also [18] where pathwise uniqueness is proved for a perturbed Tanaka’s SDE). It should be generalizable to various situations. The first SDE represents the motion of particles driven by two independent white noises W + and W −. All particles on the positive half-line are driven by W + and therefore move parallel until they hit 0. W − drives in the same way the particles while they are on the negative side. What should happen at the origin is a priori not clear, but we should assume particles do not spend a positive measure of time there. The SDE can therefore be written = + + − dXt 1{Xt >0}W (dt) 1{Xt <0}W (dt) and will be shown to have a strong, i.e. σ(W+,W−) (Wiener) measurable, solution which is a coalescing flow of continuous maps. This is the only solution in any reasonable sense, even if one allows an extension of the probability space, i.e. other sources of randomness than the driving noises, to define the flow. If we compare this result with the one obtained in [14] for Tanaka’s equation, in which the construction of the coalescing flow requires an additional countable family of independent Bernoulli variable attached to local minima of the noise W , the Wiener measurability may seem somewhat surprising. A possible intuitive interpretation is the following: In the case of Tanaka’s equation, if we consider the image of zero, a choice has to be made at the beginning of every excursion outside of zero of the driving reflected Brownian motion. In the case of our SDE, an analogous role is played by excursions of W + and W −. These excursions have to be taken at various levels different of 0, but the essential point is that at given levels they a.s. never start at the same time. And if we could a priori neglect the effect of the excursions of height smaller than some positive ε, the motion of a particle starting at zero would be perfectly determined. The second SDE is a transport equation, which cannot be induced by a flow of maps. The matter is dispersed according to the heat equation on the negative half line and is driven by W + on the positive half line. A solution is easily constructed by integrating out W − in the solution of our first equation. We will prove that also in this case, there is no other solution, even on an extended probability space. The third flow is not related to an SDE. It is constructed in a similar way as Arratia flow (see [1,5,6,12,20]) is constructed: the n-point motion is given by independent Brownian motions that coalesce when they meet (without this condition the matter is dispersed according to the heat equation on the line). Using the same procedure, each particle is driven on the negative half line by an independent white noises and on the positive half line by W +.This procedure allows to define a coalescing flow of maps, which is not Wiener measurable (i.e. not σ(W+)-measurable). 2. Notation, definitions and results 2.1. Notation + n + n • For n ≥ 1, C(R : R ) (resp. Cb(R : R )) denotes the space of continuous (resp. bounded continuous) functions f : R+ → Rn. n n • For n ≥ 1, C0(R ) is the space of continuous functions f : R → R converging to 0 at infinity. It is equipped with = | | the norm f ∞ supx∈Rn f(x). • ≥ 2 Rn : Rn → R For n 1, C0 ( ) is the space of twice continuously differentiable functions f converging to 0 at infinity = + + as well as their derivatives. It is equipped with the norm f 2,∞ f ∞ i ∂if ∞ i,j ∂i ∂j f ∞. • For a metric space M, B(M) denotes the Borel σ -field on M. n n • For n ≥ 1, M(R ) (resp. Mb(R )) denotes the space of measurable (resp. bounded measurable) functions f : Rn → R. • We denote by F the space M(R). It will be equipped with the σ -field generated by f → f(x)for all x ∈ R. • P(R) denotes the space of probability measures on (R, B(R)). The spaceP(R) is equipped with the topology of narrow convergence. For f ∈ Mb(R) and μ ∈ P(R), μf or μ(f ) denotes R f dμ = R f(x)μ(dx). Three examples of Brownian flows on R 1325 • A kernel is a measurable function K from R into P(R). Denote by E the space of all kernels on R.For f ∈ Mb(R), Kf ∈ Mb(R) is defined by Kf (x) = R f(y)K(x,dy).Forμ ∈ P(R), μK ∈ P(R) is defined by (μK)f = μ(Kf ).IfK1 and K2 are two kernels then K1K2 is the kernel defined by (K1K2)f (x) = K1(K2f )(x)(= f(z)K1(x, dy)K2(y, dz)). The space E will be equipped with E the σ -field generated by the mappings K → μK, for every μ ∈ P(R). • (n) ≥ R Rn We denote by (resp. for n 1) the Laplacian on (resp. on ), acting on twice differentiable functions f R Rn = (n) = n ∂2 on (resp. on ) and defined by f f (resp. f i=1 2 f ). ∂xi 2.2. Definitions: Stochastic flows and n-point motions Definition 2.1. A measurable stochastic flow of mappings (SFM) ϕ on R, defined on a probability space (Ω, A, P), is a family (ϕs,t)s (1) For all s (5) For all f ∈ C0(R), x ∈ R, s 2 (6) For all s Definition 2.2. A measurable stochastic flow of kernels (SFK) K on R, defined on a probability space (Ω, A, P), is a family (Ks,t)s (1) For all s (5) For all f ∈ C0(R), x ∈ R, s 2 (6) For all s E F The law of a SFK (resp. of a SFM) is a probability measure on ( s (n) ≥ Rn Rn Definition 2.3. Let (Pt ,n 1) be a family of Feller semigroups, respectively defined on and acting on C0( ) as well as on bounded continuous functions. We say that this family is consistent as soon as n (1) for all n ≥ 1, all permutation σ of {1,...,n} and all f ∈ C0(R ), (n) σ = (n) σ Pt f Pt f , σ = where f (x1,...,xn) f(xσ1 ,...,xσn ); 1326 Y. Le Jan and O. Raimond k (2) for all k ≤ n and all f ∈ C0(R ), we have (n) ◦ = (k) ◦ Pt (f πk,n) Pt f πk,n, n k where πk,n : R → R is such that πk,n(x) = (x1,...,xk), ≤ ≤ P(n) (n) such that xi yi for i n. We will denote by x the law of the Markov process associated with Pt and starting from x ∈ Rn. This Markov process will be called the n-point motion of this family of semigroups. A general result (see Theorem 2.1 in [12]) states that there is a one to one correspondence between laws of SFK K (n) (n) (n) = [ ⊗n] and consistent family of Feller semigroups P , with the semigroup P defined by Pt E K0,t ). It can be viewed as a generalization of De Finetti’s Theorem (see [20]). 2.3. Definition: White noises R × R → R R ≥ Let C : be a covariance function on ,i.e.C is symmetric and i,j λiλj C(xi,xj ) 0 for all finite family ∈ R2 (λi,xi) . Assuming that the reproducing Hilbert space HC associated to C is separable, there exists (ei)i∈I (with = I at most countable) an orthonormal basis of HC such that C(x,y) i∈I ei(x)ei(y). Definition 2.4. A white noise of covariance C is a centered Gaussian family of real random variables (Ws,t(x), s < t,x ∈ R), such that E Ws,t(x)Wu,v(y) = [s,t]∩[u, v] × C(x,y). A standard white noise is a centerd Gaussian family of real random variables (Ws,t,s E[Ws,tWu,v]= [s,t]∩[u, v] . i Starting with (W )i∈I independent standard white noises, one can define W = (Ws,t)s = i Ws,t(x) Ws,tei(x), i∈I which is well defined in L2. i i = Although Ws,t doesn’t belong to HC , one can recover Ws,t out of W by Ws,t Ws,t,ei . Indeed, for any given i, n n n n 2 e is the limit as n →∞in H of e = λ C n where for all n, (λ ,x ) is a finite family in R . Note that i C i k k xk k k en − em 2 = λnλmC xn,xm . i i HC k k k, n,i = n n Denote Ws,t k λk Ws,t(xk ). Then for n and m, 2 2 E W n,i − W m,i = (t − s) en − em . s,t s,t i i HC i 2 n,i = i Thus one can define Ws,t as the limit in L of Ws,t . Then one easily checks that Ws,t(x) i Ws,t(x)ei(x) a.s. ≤ F K F ϕ F W For K a SFK (resp. ϕ a SFM or W a white noise), we denote for all s t by s,t (resp. s,t or s,t)theσ -field generated by {Ku,v; s ≤ u ≤ v ≤ t} (resp. by {ϕu,v; s ≤ u ≤ v ≤ t} or {Wu,v; s ≤ u ≤ v ≤ t}). A white noise W is said FK F ϕ F W ⊂ F K F W ⊂ F ϕ to be a (resp. -white noise) if s,t s,t for all s 1 2.4. Definition: The ( 2 ,C)-SDE Let ϕ be a SFM and W a Fϕ-white noise of covariance C. Then if for all s ∈ 2 R ∈ R F K for all f C0 ( ), s = 1 (2) (2) where A 2 is the generator of Pt and A is the generator of Pt .) Having C stricly less than one means the flow is a mixture of stochastic transport and deterministic heat flow. Note also that in this case there are no SFM’s solution 1 of the ( 2 ,C)-SDE. 1 Taking the expectation in (2), we see that for a solution (K, W) of the ( 2 ,C)-SDE, the one-point motion of K is a standard Brownian motion. Note that if K is a SFK of the form δϕ, with ϕ a SFM, then (K, W) is a solution of the 1 = ∈ R ( 2 ,C)-SDE if and only if (ϕ, W) solves (1), and we must have C(x,x) 1 for all x . In the following we will make the assumption that C(x,x) ≤ 1 for all x ∈ R. 1 FK = F W A solution (K, W) of the ( 2 ,C)-SDE is called a Wiener solution if for all s 1 2.5. Martingale problems related to the ( 2 ,C)-SDE 1 (n) Let (K, W) be a solution of the ( 2 ,C)-SDE. Denote by Pt , the semigroup associated with the n-point motion of (n) n K, and by Px the law of the n-point motion started from x ∈ R . (n) (n) n Proposition 2.5. Let X be distributed like Px with x ∈ R . Then it is a solution of the martingale problem: t (n) − (n) (n) f Xt A f Xs ds (3) 0 1328 Y. Le Jan and O. Raimond ∈ 2 Rn is a martingale for all f C0 ( ), where 1 ∂2 A(n)f(x)= (n)f(x)+ C(x ,x ) f(x). 2 i j ∂x ∂x i ⊗···⊗ 2 R Proof. When f is of the form f1 fn, with f1,...,fn in C0 ( ), then it is easy to verify (3) when (K, W) 2 Rn solves (2). This extends to the linear space spanned by such functions, and by density, to C0 ( ). (A proof of this fact can be derived from the observation that, on the torus, Sobolev theorem shows that trigonometric polynomials are dense in C2.) Remark 2.6. If one can prove uniqueness for these martingale problems, then this implies there exists at most one 1 solution to the ( 2 ,C)-SDE.(Indeed, using Itô’s formula, the expectation of any product of Ks,tf(x)and Ws,t(y)’s (n) can be expressed in terms of Pt ’s.) Remark 2.7. When C is smooth, then the martingale problem (3) is well posed. This implies that the n-point motion 1 of K is uniquely determined and that the ( 2 ,C)-SDE has a unique solution, which is a Wiener solution. If one assumes in addition that C(x,x) = 1, then the usual theory applies, and the solution is a flow of diffeomorphisms. 2.6. Statement of the main theorems = + 1 Theorem 2.8. Let C±(x, y) 1{x>0}1{y>0} 1{x<0}1{y<0}. Then there is a unique solution (K, W) of the ( 2 ,C±)- SDE. Moreover, this solution is a Wiener solution and K is a coalescing SFM. ± Denote by ϕ the coalescing flow defined in Theorem 2.8. The white noise W of covariance C± can be written in + − + − the form W = W 1R+ + W 1R− , with W and W two independent standard white noises. Then (2) is equivalent to t t ± + − ϕ (x) = x + 1 ± W (du) + 1 ± W (du). (4) s,t {ϕs,u(x)>0} {ϕs,u(x)<0} s s A consequence of this theorem, with Proposition 4.1 below, is Theorem 2.9 below. In this theorem, the solutions of the SDE (5) are in the usual sense, that is they are single paths. After completing this work, we were informed by H. Hajri that a result proved in [18] and in [2] implies that pathwise uniqueness holds for this SDE. Theorem 2.9. The SDE driven by B+ and B−, two independent Brownian motions, = + + − dXt 1{Xt >0} dBt 1{Xt <0} dBt (5) has a unique solution. Moreover, this solution is a strong solution. Proof. We recall that saying (5) has a unique solution means that for all x ∈ R, there exists one and only one proba- + 3 + − bility measure Qx on C(R : R ) such that under Qx(dω), the canonical process (X, B ,B )(ω) = ω satisfies (5), with B+ and B− two independent Brownian motions, and X a Brownian motion started at x. Proposition 4.1 states + − = ± + − that (5) has a unique solution. Since (4) holds, one can take (Xt ,Bt ,Bt ) (ϕ0,t (x), W0,t ,W0,t ) for this solution. A solution (X, B+,B−) is a strong solution if X is measurable with respect to the σ -field generated by B+ and B−, completed by the events of probability 0. Since ϕ± is a Wiener solution, one can conclude. Theorem 2.10. Let C+(x, y) = 1{x>0}1{y>0}. + 1 (i) There is a unique solution (K ,W)solution of the ( 2 ,C+)-SDE. (ii) The flow K+ is diffusive and is a Wiener solution. + ± + + + (iii) The flow K can be obtained by filtering ϕ with respect to the noise generated by W : K = E[δ ± |W ]. s,t ϕs,t Three examples of Brownian flows on R 1329 Theorem 2.11. There exists a unique coalescing SFM ϕ+ such that its n-point motions coincide with the n-point + n motions of K before hitting n ={x ∈ R ;∃i = j,xi = xj }. Moreover + + 1 (i) (ϕ ,W ) is not a solution of the ( 2 ,C±)-SDE but it satisfies t t + + 1 + dϕ (x) = 1 + W (du). {ϕs,u(x)>0} s,u {ϕs,u(x)>0} s s + + + + + (ii) K can be obtained by filtering ϕ with respect to the noise generated by W : K = E[δ + |W ]. s,t ϕs,t Remark 2.12. Following [12], it should be possible to prove that he linear part of the noise generated by ϕ+ is the noise generated by W +. We refer to Section 3.2 for more precise definitions of noises, extension of noises, filtering by a subnoise, and linear part of a noise. 3. General results In this section C is any covariance function on R satisfying C(x,x) ≤ 1 for all x ∈ R. 3.1. Chaos decomposition of Wiener solutions 1 Proposition 3.1. Let (K, W) be a Wiener solution of the ( 2 ,C)-SDE. Then for all s = n Ks,tf(x) Js,tf(x) (6) n≥0 n 0 = R ≥ with J defined by Jt Pt , where Pt is the heat semigroup on , and for n 0, t n+1 = n i Js,t f(x) Js,u (Pt−uf)ei (x)W (du). (7) i s 1 This implies that there exists at most one Wiener solution to the ( 2 ,C)-SDE. Proof. We essentially follow the proof of Theorem 3.2 in [11], with a minor correction at the end noticed by Bertrand Micaux during his Ph.D. [15]. t i 2 Let us first remark that the stochastic integral i 0 Ks(ei(Pt−sf))(x)W (ds) do converge in L for all bounded measurable function f since (using in the fourth inequality that C(x,x) ≤ 1 for all x ∈ R) t 2 i E Ks,u (Pt−uf)ei (x)W (du) i s t 2 ≤ E Ks,u (Pt−uf)ei (x) du i s t 2 ≤ E Ks,u (Pt−uf)ei (x) du i s t ≤ 2 2 Pu−s (Pt−uf) ei (x) du s i t 2 ≤ Pu−s (Pt−uf) (x) du s 1330 Y. Le Jan and O. Raimond d 2 = 2 2 − 2 which is finite (since, using that du (Pu−s(Pt−uf) ) Pu−s((Pt−uf)) , it is equal to Pt−sf (x) (Pt−sf) (x)). 3 Take f a C function with compact support. For t>0, denote K0,t simply Kt . Then for t>0, n ≥ 1 and x ∈ R n−1 Kt f(x)− Pt f(x)= Kt(k+1)/n(Pt(1−(k+1)/n)f)− Ktk/n(Pt(1−k/n)f) k=0 n−1 = (Kt(k+1)/n − Ktk/n)(Pt(1−(k+1)/n)f) k=0 n−1 + Ktk/n(Pt(1−(k+1)/n)f − Pt(1−k/n)f) k=0 n−1 t(k+1)/n i = Ks ei(Pt(1−(k+1)/n)f) (x)W (ds) k=0 i tk/n n−1 t(k+1)/n 1 + K (P − + f) (x) ds s 2 t(1 (k 1)/n) k=0 tk/n n−1 − Ktk/n(Pt(1−k/n)f − Pt(1−(k+1)/n)f) k=0 1 ≥ ∈ R since (K, W) solves the ( 2 ,C)-SDE. This last expression implies that for t>0, n 1 and x , t 3 i Kt f(x)− Pt f(x)− Ks ei(Pt−sf) (x)W (ds) = Bk(n) i 0 k=1 with n−1 t(k+1)/n i B1(n) = Ks ei(Pt(1−(k+1)/n)f − Pt−sf) (x)W (ds), k=0 i tk/n n−1 t B (n) =− K P − f − P − + f − (P − + f) (x), 2 tk/n t(1 k/n) t(1 (k 1)/n) 2n t(1 (k 1)/n) k=0 n−1 t(k+1)/n 1 B (n) = (K − K ) (P − + f) (x) ds. 3 s tk/n 2 t(1 (k 1)/n) k=0 tk/n The terms in the expression of B1(n) being orthogonal, n−1 t(k+1)/n 2 2 E B1(n) = E Ks ei(Pt(1−(k+1)/n)f − Pt−sf) (x) ds k=0 i tk/n n−1 t(k+1)/n 2 ≤ Ps ei(Pt(1−(k+1)/n)f − Pt−sf) (x) ds k=0 i tk/n n−1 t(k+1)/n 2 ≤ Ps (Pt(1−(k+1)/n)f − Pt−sf) (x) ds, k=0 tk/n Three examples of Brownian flows on R 1331 2 = ≤ where we have used Jensen inequality in the second inequality and the fact that i ei (x) C(x,x) 1inthelast t/n − 2 = 2 2 inequality. This last term is less that n 0 Psf f ∞ ds O( f ∞t /n). 2 1/2 By using triangular inequality, E[(B2(n)) ] is less than n−1 2 1/2 t E K P − f − P − + f − (P − + f) (x) tk/n t(1 k/n) t(1 (k 1)/n) 2n t(1 (k 1)/n) k=0 n−1 2 1/2 t ≤ P P − f − P − + f − (P − + f) (x) . tk/n t(1 k/n) t(1 (k 1)/n) 2n t(1 (k 1)/n) k=0 3/2 2 1/2 Using moreover that Pt/nf − f − t/(2n)f ∞ = O((t/n) f ∞), we get that E[(B2(n)) ] = − O(t3/2n 1/2f ∞). 2 1/2 By using again triangular inequality, E[(B3(n)) ] is less than n−1 t(k+1)/n 2 1/2 1 E (K − K ) (P − + f) (x) ds s tk/n 2 t(1 (k 1)/n) k=0 tk/n n−1 t(k+1)/n 2 1/2 t 1 ≤ E (K − K ) (P − + f) (x) ds . n s tk/n 2 t(1 (k 1)/n) k=0 tk/n ≤ P(2) (2) For f a Lipschitz function and 0 s with Lip(f ) the Lipschitz constant of f . From this estimate, and since Lip(Pt(1−(k+1)/n)f ) ≤ Lip(f ), we deduce 2 1/2 3/2 −1/2 2 that E[(B3(n)) ] = O(t Lip(f )n ). The estimates we gave for E[(Bi(n)) ],fori ∈{1, 2, 3}, implies that for f a C3 function with compact support, t i Kt f(x)= Pt f(x)+ Ks ei(Pt−sf) (x)W (ds). i 0 This implies that t i Ks,tf(x)= Pt−sf(x)+ Ks,u (Pt−uf) (x)W (du). (8) i s Iterating n times relation (8), we get that n = k + n Ks,tf(x) Js,tf(x) Rs,tf(x), k=0 n 2 where Rs,tf(x) is a L random variable whose chaos expansion is such that its n first Wiener chaoses are 0. This k 3 implies that for all k, Js,tf(x)is the kth Wiener chaos of Ks,tf(x). Thus, for f a C function with compact support, the Wiener chaos expansion of Ks,tf(x)is given by (6). This extends to all bounded measurable functions. 1332 Y. Le Jan and O. Raimond 3.2. Filtering a SFK by a subnoise We follow here Section 3 in [12]. Definition 3.2. A noise consists of a separable probability space (Ω, A, P), a one-parameter group (Th)h∈R of P- 2 preserving L -continuous transformations of Ω and a family Fs,t, ≤ s ≤ t ≤∞of sub-σ -fields of A such that: (a) Th sends Fs,t onto Fs+h,t+h for all h ∈ R and s ≤ t, (b) Fs,t and Ft,u are independent for all s ≤ t ≤ u, (c) Fs,t ∧ Ft,u = Fs,u for all s ≤ t ≤ u. Moreover, we will assume that, for all s ≤ t, Fs,t contains all P-negligible sets of F−∞,∞, denoted F. ¯ ¯ ¯ A subnoise N of N is a noise (Ω, A, P,(Th), Fs,t), with Fs,t ⊂ Fs,t. Note that a subnoise is characterized by a ¯ ¯ ¯ ¯ σ -field invariant by Th for all h ∈ R, F = F−∞,∞. In this case, we will say that N is the noise generated by F. A linear representation of N is a family of real random variables X = (Xs,t; s ≤ t) such that: (a) Xs,t ◦ Th = Xs+h,t+h for all s ≤ t and h ∈ R, (b) Xs,t is Fs,t-measurable for all s ≤ t, (c) Xr,s + Xs,t = Xr,t a.s., for all r ≤ s ≤ t. lin Define F be the σ -field generated by the random variables Xs,t where X is a linear representation of N and s ≤ t. lin lin Define N to be the subnoise of N generated by F . This noise is called the linear part of the noise N. 0 0 A0 = E Let P be the law of a SFK, it is a law on (Ω , ) ( s P 1 = Suppose now 0 is the law of a SFK that solves the ( 2 ,C)-SDE. Writing C in the form C(x,y) F K = i i∈I ei(x)ei(y), there is a -white noise W i∈I W ei such that: t t 1 K f(x)= f(x)+ K e f (x)W i(du) + K f (x) du (9) s,t s,u i 2 s,u i∈I s s 1 (i.e. the ( 2 ,C)-SDE is transported on the canonical space). ⊂ ¯ { j ; ∈ } ¯ = j ¯ Proposition 3.3. Let J I and N the noise generated by W j J , and W j∈J W ej . Let K be the SFK ¯ ¯ ¯ obtained by filtering K with respect to N. Then (9) is satisfied with K replaced by K and W by W (or I by J ). If ¯ ⊂ ¯ ¯ 1 ¯ ¯ = ¯ σ(W) σ(K), then K solves the ( 2 ,C)-SDE with C(x,y) j∈J ej (x)ej (y), and driven by W . Proof. This proposition reduces to prove that t t 1 K¯ f(x)= f(x)+ K¯ e f (x)W j (du) + K¯ f (x) du s,t s,u j 2 s,u j∈J s s which easily follows by taking the conditional expectation with respect to σ(Wj ; j ∈ J)in equation (9). Three examples of Brownian flows on R 1333 4. C±(x, y) = 1{x<0}1{y<0} + 1{x>0}1{y>0} In this section, we will prove Theorem 2.8. In order to do this we will prove that the n-point motion of a solution is uniquely determined and has the coalescing property (when two points meet, they stay together). In Section 4.2, we prove that such a n-point motion is uniquely determined up two the first time to of them meet. In Section 4.3, we show that the process obtained from the two-point motion after deleting its excursions out of D ={(x, y) ∈ R2; xy ≤ 0}, and stopped when it reaches (0, 0) is a Brownian motion with oblique reflection at the boundary of D, stopped when it reaches (0, 0). In Section 4.4, after having shown that the two-point motion of a solution reaches (0, 0) in finite time, we construct a consistent family of n-point motions verifying the coalescing property. This allows to construct a coalescing SFM 1 and a solution to the ( 2 ,C±)-SDE. In Section 4.5, we show that any two point-motion of a solution cannot leave {(x, y) ∈ R2; x = y}. This is done by studying the process obtained from the two-point motion after deleting its excursions out of D. = + + − 4.1. The SDE dXt 1{Xt >0} dBt 1{Xt <0} dBt Let B1 and B2 be two independent Brownian motions. For x ∈ R, define Xx , Bx,+ and Bx,− by x = + 1 Xt x Bt , (10) t t x,− x 2 = { x } + { x } Bt 1 Xs <0 dXs 1 Xs >0 dBs , (11) 0 0 t t x,+ 2 x = { x } + { x } Bt 1 Xs <0 dBs 1 Xs >0 dXs . (12) 0 0 Then Xx , Bx,− and Bx,+ are Brownian motions respectively started at x, 0 and 0. Moreover Bx,− and Bx,+ are independent, and we have t t x x,+ x,− = + { x } + { x } Xt x 1 Xs >0 dBs 1 Xs <0 dBs . (13) 0 0 x x,− x,+ Denote by Qx the law of (X ,B ,B ). Proposition 4.1. Let x ∈ R, X, B+ and B− be real random processes such that B+ and B− are independent Brown- ian motions. Then if t t = + + + − Xt x 1{Xs >0} dBs 1{Xs <0} dBs , (14) 0 0 + − the process (X, B ,B ) has for law Qx . Proof. Assume (X, B+,B−) satisfy (14), with B+ and B− two independent Brownian motions. Let t = + + − Bt 1{Xs <0} dBs 1{Xs >0} dBs . 0 Observe that X − x and B are two independent Brownian motions. Moreover t t − = + Bt 1{Xs <0} dXs 1{Xs >0} dBs, (15) 0 0 t t + = + Bt 1{Xs <0} dBs 1{Xs >0} dXs. (16) 0 0 Note that (X, B+,B−) is defined out of (X − x,B) exactly in the same way as (Xx,Bx,+,Bx,−) is defined out of (B1,B2). This implies the proposition ((X, B+,B−) is distributed like (Xx,Bx,+,Bx,−)). 1334 Y. Le Jan and O. Raimond 4.2. Construction of the n-point motions up to T (n) n Let x/∈ n ={x ∈ R ;∃i = j,xi = xj }. For convenience, we will assume that x1 0 = + 1 Xi (t) xi Bt , t t 0,− = 1 + 2 Bt 1{X0(s)<0} dBs 1{X0(s)>0} dBs , 0 i 0 i t t 0,+ = 2 + 1 Bt 1{X0(s)<0} dBs 1{X0(s)>0} dBs , 0 i 0 i 0 = + 0,− ≤ − Xj (t) xj Bt for j i 1, 0 = + 0,+ ≥ + Xj (t) xj Bt for j i 1. Set = 0 = 0 = τ1 inf t>0: Xi−1(t) 0orXi+1(t) 0 − + 0 0,− 0,+ and set for t ≤ τ1, (X, B ,B )(t) = (X ,B ,B )(t). Set i1 = i + 1ifXi+1(τ1) = 0 and i1 = i − 1if Xi−1(τ1) = 0. ··· = ··· Then X1(τ1) • (τk)1≤k≤ is an increasing sequence of stopping times with respect to the filtration associated to X; • X1(τk)<··· (n) (n) = (n) = { ; (n) ∈ } Lemma 4.2. Let X be a solution to the martingale problem (3), with X0 x. Let T inf t Xt n) . Then (n) P(n),0 (Xt )t≤T (n) is distributed like x . n Proof. Let x = (x1,...,xn) ∈ R be such that x1 < ··· Let B0 be a Brownian motion, independent of X(n). Set = ∃ = (n) = τ1 inf t>0: j i, Xj (t) 0 . Three examples of Brownian flows on R 1335 Define for t ≤ τ1, + = (n) − ≤ − Bt Xi+1(t) xi+1, when i n 1, t t + (n) 0 B = (n) X (s) + (n) B , i = n. t 1{X (s)>0} d i 1{X (s)<0} d s when 0 i 0 i Define also for t ≤ τ1 − = (n) − ≥ Bt Xi−1(t) xi−1, when i 2, t t − (n) 0 B = (n) X (s) + (n) B (s), i = . t 1{X (s)<0} d i 1{X (s)>0} d when 1 0 i 0 i Define for t ≤ τ1, 1 = (n) − Bt Xi (t) xi and t t 2 − + B = (n) B + (n) B . t 1{X (s)>0} d s 1{X (s)<0} d s 0 i 0 i 1 2 Note that for t ≤ τ1, B ,B t = 0 and that (n) = + − Xj (t) xj Bt for j 1 2 + − Assume now that (τk)k≤ and (Bt ,Bt ,Bt ,Bt )t≤τ have been defined such that a.s. (n) • (τk)1≤k≤ is an increasing sequence of stopping times with respect to the filtration associated to X ; • (n) ∈ ≤ ≤ Xτk / n for 1 k ; • for all 1 ≤ k ≤ , there exists an integer i such that X(n)(τ ) = 0. k ik k 1 2 + − 1 2 + − We then define τl+1 and (Bt ,Bt ,Bt ,Bt )τ 4.3. Brownian motions with oblique reflection Let X and B be two independent Brownian motions, with X started at x ∈ R.LetY be a Brownian motion started at y defined by t t = + + Yt y 1{Xs <0} dBs 1{Xs >0} dXs. 0 0 = t = { ; } r r = Let At 0 1{Xs <0} ds and κt inf s>0 As >t . Then define (X ,Y )t (X, Y )κt , it is a continuous process r − taking its values in {x ≤ 0}. Denote by Lt the local time at 0 of Xt . Then X is a Brownian motion in R reflected at r = 1 1 = r − r − r 0 and that if Lt 2 Lκt , Bt Xt Lt X0 is a Brownian motion. 2 = κt 2 1 Lemma 4.3. Set Bt 0 1{Xs <0} dBs . Then B is a Brownian motion independent of B and we have r = 2 − r + r Yt Bt Lt Y0 . 1336 Y. Le Jan and O. Raimond Thus (Xr ,Yr ) is a Brownian motion in {x ≤ 0} reflected at {x = 0} with oblique reflection with angle of reflection equal to π/4, i.e. r r = r r + 1 2 − r r Xt ,Yt X0,Y0 Bt ,Bt Lt ,Lt r r = ≤ r r = − (note that (X0,Y0 ) (x, y) if x 0 and (X0,Y0 ) (0,y x) if x>0). 1 = κt 1 = 2 = κt 1 2 = 1 2 Proof. Note that Bt 0 1{Xs <0} dXs . Since B t B t 0 1{Xs <0} ds and since B ,B t 0, B and B are two independent Brownian motions. Let ε be a small positive parameter and define the sequences of stopping ε ε ε = ≥ times σk and τk such that σ0 0 and for k 0 ε ε τ = inf t ≥ σ ; Xt = 0 , k k ε = ≥ ε; = σk+1 inf t τk Xt ε . r = ≤ r = Note that X0 x if x 0 and X0 0ifx>0, and for t>0 r X = (X ε∧ − X ε ∧ ) + X ε∧ , t τk κt τk−1 κt τ0 κt k≥1 ε ε r with X ε∧ = X if x ≤ 0 and κ <τ and X ε∧ = 0ifκ >τ (which holds if x>0). We have that X = τ0 κt t t 0 τ0 κt t 0 t − ε,r + ε,1 + r Lt Bt X0 with ε,r L =− (X ε∧ − X ε∧ ), t τk κt σk κt k≥1 ε,1 r B = (X ε∧ − X ε ∧ ) + X ε∧ − X . t σk κt τk−1 κt τ0 κt 0 k≥1 ε,r ε,1 r 1 Then Lt and Bt both converges in probability respectively towards Lt and Bt (see Theorem 2.23 in Chapter 6 of [10]). r = ≤ r = − Note now that Y0 y is x 0 and Y0 y x if x>0, and for t>0 r Y = (Y ε∧ − Y ε ∧ ) + Y ε∧ , t τk κt τk−1 κt τ0 κt k≥1 ε ε with Y ε∧ = Y if x ≤ 0 and κ <τ and Y ε∧ = y − x if κ >τ (which holds if x>0). Since when X is positive, τ0 κt t t 0 τ0 κt t 0 t − r =− ε,r + ε,2 + r Xt Yt remains constant, Yt Lt Bt Y0 , with ε,2 r B = (Y ε∧ − Y ε ∧ ) + Y ε∧ − Y . t σk κt τk−1 κt τ0 κt 0 k≥1 ε,2 2 It remains to observe that Bt converges in probability towards Bt . Denote now by (X, Y ) a solution of the martingale problem (3), for n = 2. Set T = inf{s ≥ 0; Xs = Ys}. + ={ ∈ R2; ≤ ≥ } − ={ ∈ R2; ≥ ≤ } = + ∪ − ± = Let D (x, y) x 0 and y 0 , D (x, y) x 0 and y 0 and D D D .LetAt t∧T + − ± ± ± ± = + = { ; } = { ; } ≤ 0 1{(Xs ,Ys )∈D } ds and At At At .Letκt inf s As >t and κt inf s As >t . Set for t AT , r,± r,± = ± ≤ r r = (Xt ,Yt ) (X, Y )(κt ) and for t AT , (Xt ,Yt ) (X, Y )(κt ). Note that if X0 >Y0 (resp. if X0 Lemma 4.4. The process (Xr ,Yr ) is a Brownian motion in D, with oblique reflection at the boundary of angle of reflection equal to π/4, and stopped when it hits (0, 0). More precisely, there exists B1 and B2 two independent Brownian motions such that for all t ≤ AT , when X0 This process is a special case of the class of processes studied in [21,22]: the process (Xr ,Yr ) is a Brownian motion in the wedge D, with angle ξ = π/2 and angles of reflection θ1 = θ2 = π/4. An important parameter in the study of these processes is α = (θ1 + θ2)/ξ = 1. The fact that the reflected Brownian motion is a semimartingale on [0,τ0], where τ0 is the first time to hit the origin, follows from Theorem 1 of [22]. An alternative proof, in a more general multi-dimensional and state-dependent coefficient setting, is given in Theorem 1.4 (property 3) of [19]. The non-semimartingale property of the Markovian extension spending no time at zero follows from Theorem 5 (with α = 1) of [22]. Two alternative proofs, the first of which also generalizes to more general diffusions and higher dimensions, are given in Theorem 3.1 of [9] and Proposition 4.13 of [3]. These results are essentially related to the study of the local times which is done in the next section. We choose not to derive the results of this study from these references in order to keep their proofs self contained and reasonably short. 4.4. Coalescing n-point motions r + r r − r r r + Yt √ Xt Yt √ Xt Without loss of generality, assume that (X ,Y ) ∈ D .LetUt = and Vt = . Then V is a Brownian 0 0 2 2 = { ≥ ; r = r = } motion stopped when it hits 0. This implies in particular that T0 inf t 0 Xt Yt 0 is finite a.s. Denote by Lr := L1 + L2 the local time at 0 of (Xr ,Yr ) at the boundary {x = 0}∪{y = 0}. Lemma 4.5. P(Lr < ∞) = 1. T0 = { ≥ ; = } Proof. Fix M>0 and denote TM inf t 0 Vt M . Note that the process (Ut ,Vt )t≤T0 is a Brownian motion with oblique reflection in {−v ≤ u ≤ v}, stopped when it hits (0, 0).Fort 2+ 1 2− 1 2+ 2 where W 1 = B √ B and W 2 = B √ B are two independent Brownian motions. Let h(u, v) = u v . Then, when 2 2 2v u =±v, ∂vh(u, v) = 0 and ∂uh(u, v) =±1. Note also that in {−v u2 + v2 2 h(u, v) = ≤ . v3 v Applying Itô’s formula, we get for t Since this quadratic variation is dominated by t, (Mt )t≤T0 is a martingale. This implies that t∧T0∧TM √ ≤ + ds − [ ] E h (U, V )t∧T0∧TM E h(U0,V0) E 2E Lt∧T0∧TM . 0 Vs T ∧T It is well known that E[ 0 M ds ] < ∞. For example, using Itô’s formula we get for t We then must have that E[Lr ] < ∞. So a.s., Lr < ∞ for all M>0. This proves the lemma. T0∧TM T0∧TM Lemma 4.6. Consider again (Xt ,Yt ) a solution of the martingale problem (3) for n = 2. Then T = inf{t ≥ 0; Xt = Yt } is finite a.s. r r ∈ + ∈ Proof. Suppose Y0 >X0, so that (X0,Y0 ) D . To simplify a little, we will also assume that (X0,Y0) D.Note first that when T<∞,wemusthaveXT = YT = 0. Since we are only interested to (X, Y ) up to T , when T<∞ we will replace (XT +t ,YT +t )t>0 by (Bt ,Bt )t>0, with B a Brownian motion independent of (Xt ,Yt ){t≤T }.Fixε>0 and ε ε ε = ≥ define the sequences of stopping times σk and τk ,byτ0 0 and for k 1 ε = ≥ ε = =− σk inf t τk−1,Xt ε or Yt ε , ε = ≥ ε = = τk inf t σk ,Xt 0orYt 0 . ∈ [ ε ε] ε ∧ − ε ∧ Then all these stopping times are finite a.s. Let us also remark that T/ k σk ,τk and that k(τk T σk T) c converges a.s. towards T − T (i.e. the time spent by (X, Y ) in D before T ). Take α>0. Denoting Z = (X, Y ), Z ε 0 τk is a Markov chain. Note that Nε = inf{k,Zτ ε = (0, 0)} is a stopping time for this Markov chain. Moreover it is clear ε ε k that (τ − σ ) ≥ is a sequence of independent variables, and independent of the Markov chain Z ε . And we have for k k k 1 √ τk −α(τε−σ ε) − all k ≥ 1, E[e k k ]=e ε 2α. Hence ε− − α(τε∧T −σ ε∧T) − N 1 α(τε−σ ε) E e k k k = E e k=1 k k √ −(n−1)ε 2α = P[Nε = n]e n which implies that √ −α (τ ε∧T −σ ε∧T) −(N −1)ε 2α E e k k k = E e ε . By taking the limit as ε → 0 in this equality, since εNε converges in probability towards LT (X) + LT (Y ). Denote + L := LT (X) LT (Y ) . Note that L = Lr , so that we have T 2 T T0 √ r −α(T −T ) −2 2αL E e 0 = E e T0 . Three examples of Brownian flows on R 1339 Since Lr is finite a.s., T − T is also finite a.s. T0 0 (n) ≥ R (n) Proposition 4.7. There exists a unique consistent family (Pt ,n 1) of Feller semigroups on such that if X is ∈ Rn (n) = { ≥ ; (n) ∈ } its associated n-point motion started from x and T inf t 0 Xt n , then: (n) P(n),0 (i) (Xt )t≤T (n) is distributed like x . ≥ (n) (n) ∈ (ii) For t T , Xt n. Moreover, this family is associated to a coalescing SFM. Proof. To prove the Feller property, it suffices to check condition (C) of Theorem 4.1 in [12]. Denoting by (X, Y ) the two-point motion, condition (C) is verified as soon as for all positive t and ε (denoting d(x,y) =|y − x|) (2) lim P d(Xt ,Yt )>ε = 0. (18) d(x,y)→0 (x,y) Assume 0 + − Proposition 4.8. Let ϕ be the SFM associated to P(n). Then there exist W and W two independent white noises, σ(ϕ)-measurable, such that (4) is satisfied. ± = − Proof. Define for s 4.5. Uniqueness Let P(n) be another consistent family solving the martingale problem (3). By uniqueness of the solution of this mar- (n) tingale problem up to the first coalescing time, if this consistent family is different to the family P , then the n- t point motion should leave n. After a time change, one may assume it spends no time in n. Denote by (X,Y) t the associated two-point motion starting from (0, 0). Denote A = 1 ds, κ = inf{s ≥ 0; A ≥ t} and t 0 {(Xs ,Ys )∈D} t s r r = 1 + (Xt , Yt ) (X,Y)κt . Denote Lt (X) and Lt (Y)the local times at 0 of X and of Y . Define also Lt by 2 (Lt (X) Lt (Y)) r r r Y r√+Xr r Y r√−Xr and L by L . Denote U = and V =| |. t κt 2 2 Lemma 4.9. For all a>0, Ta = inf{t>0; d(Xt , Yt ) = a} is infinite a.s. a a ε Proof. Let T r = inf{t>0; V r = √ }. Then one has A = T r .Letε<√ . Define σ ε and τ ε by τ = 0 and for a t 2 Ta a 2 n n 0 n ≥ 1, ε = ≥ ε ; r = σn inf t τn−1 Vt ε , ε = ≥ ε; r = τn inf t σn Vt 0 . 1340 Y. Le Jan and O. Raimond = u2+v2 [ ε ∧ r ε ∧ r ] r = r r Recall h(u, v) 2v . Applying Itô’s formula on the time intervals σn Ta ,τn Ta , denoting Z (U , V ), ≥ 1 and using h(u, v) v for the second term, r − ≥ r − r h Zr h(0) h Z ε∧r h Z ε ∧r (19) Ta τn Ta σn−1 Ta n≥1 ε∧r τn Ta + 1 ds (20) ε∧r 2V r n≥1 σn Ta s − r − r L ε∧ L ε∧ (21) τn Ta σn Ta n≥1 + n,ε − n,ε M ε∧r M ε∧r , (22) τn Ta σn Ta n≥1 where for all n ≥ 1 and all ε>0, Mn,ε is a martingale whose quadratic variation is such that d Mn,ε ≤ 1. Since dt t √ T B r ds =∞a.s. (see Corollary 6.28 Chapter 3 in [10]), with T B the first time a Brownian motion B hits −a/ 2or 0 √ |Bs | r ∞ → × { ; ε ≤ r } a/ 2, we get that term (20) goes to as ε 0. Terms (19) is positive and is dominated by ε # n τn Ta which 0 0 converges in probability towards Lr , with L the local time at 0 of V , which is a reflected Brownian motion. Denote Ta by Mε the martingale defined by ε n,ε n,ε M = M ε∧ − M ε∧ . t τn t σn t n≥1 ε ε ε ε d n,ε ε ε ε r Then if ε <ε, M − M t = M t − M t . Using that M t ≤ 1, we get M − M r ≤ ≥ (σ ∧ T − dt Ta n 1 n a ε ∧ r [ ε ∧ r − ε ∧ r ] → ε τn−1 Ta ). Since E n≥1(σn Ta τn−1 Ta ) converges towards 0 as ε 0, we get that Mr is a Cauchy Ta 2 P(2) ε 2 P(2) sequence in L ( (0,0)) and thus (22), which is equal to Mr , converges in L ( (0,0)). This implies that term (21) Ta ∞ r r =∞ converges in probability towards . Since it also converges towards Lr . One concludes that Lr a.s. Since Ta Ta = r = =∞ =∞ Ta κTr , Lr LT and therefore LT a.s. Since X and Y are two Brownian motions, one must have Ta a Ta a a a.s. (2) (2) This lemma implies that after hitting 2 ={x = y}, the two-point motion stays in 2. Thus P = P .Thesame argument applies to prove that P(n) = P(n). This proves that there exists only one flow which is a SFK solution of the 1 ± ( 2 ,C±)-SDE. Moreover this solution is a SFM ϕ . To prove ϕ± is a Wiener solution, define K¯ the SFK obtained by filtering ϕ± with respect to the noise N ± generated by W + and W −. Then, like for Proposition 4.8 we have σ(W+,W−) ⊂ σ(K)¯ , and applying Proposition 3.3, K¯ also + − 1 + + − ¯ solves the ( 2 ,C±)-SDE driven by W 1R W 1R . Since there exists a unique solution, K is distributed like ¯ ± ± + − ± δϕ± . The noise of K being N , this implies that ϕ is measurable with respect to σ(W ,W ),i.e.ϕ is a Wiener solution. This concludes the proof of Theorem 2.8. 5. C+(x, y) = 1{x>0}1{y>0} In this section, we prove Theorem 2.10 and Theorem 2.11. 1 First (in Section 5.1), by filtering the solution of the ( 2 ,C±)-SDE, we construct the Wiener solution to the 1 ( 2 ,C+)-SDE. 1 In Section 5.2, we determine the n-point motion of the ( 2 ,C±)-SDE up to the first time two of them meet. In Section 5.3, associating to the family of n-point motions of the Wiener solution a consistent family of coalescing n-point motions, we construct a coalescing SFM. Using Section 5.2, this implies Theorem 2.11. Three examples of Brownian flows on R 1341 1 In Section 5.4, we prove the uniqueness of the solution of the ( 2 ,C+)-SDE. We first remark that this holds if and only if the two-point motion of a solution is the one of the Wiener solution. We then show there is at most one possible two-point motion. 5.1. The Wiener solution obtained by filtering ± 1 + + + Let ϕ be the SFM solution of the ( 2 ,C±)-SDE, and N the noise generated by W . Define K the SFK obtained ± + ∈ R ∈ R ∈[ ] = { ; + =− } by filtering ϕ with respect to N . Note that for x , s and t s,τs(x) with τs(x) inf u Ws,u x ,we ± + + + K+ have ϕ (x) = x + W and thus that K (x) = δ + + . This clearly implies that W = W 1R+ is a F -white noise. s,t s,t s,t x Ws,t + + 1 = + Then (see Section 3.2) K is a Wiener solution of the ( 2 ,C+)-SDE driven W W 1R . Proposition 3.1 implies it is the unique Wiener solution. 5.2. Construction of the n-point motions n n Let Dn ={x ∈ R ;∃i = j,xi = xj ≥ 0}.Letx ∈ R \ Dn. The n-point motion has the property that at any time, the points located on the positive half line will move par- allel to W + and the points located on the negative half line will follow independent Brownian motions. Its law is determined up to the first time it hits Dn. Indeed, denote I− ={j; xj ≤ 0} and I+ ={j; xj > 0}.Leti be such that 1 n xi = max{xj ; j ∈ I−} when I− = ∅ and xi = min{xj ; 1 ≤ j ≤ n} when I− = ∅.Let(B ,...,B ,B) be n + 1 inde- pendent Brownian motions. Define for t>0, the processes 0 = + j ∈ ∪{ } Xj (t) xj B (t) for j I− i , t t 0,+ = + i Bt 1{X0(s)<0} dBs 1{X0(s)>0} dBs , 0 i 0 i 0 = + 0,+ ∈ \{ } Xj (t) xj Bt for j I+ i . Set = ;∃ = 0 = τ1 inf t>0 j i, Xj (t) 0 ≤ = 0 + = 0,+ and set for t τ1, Xt Xt and Bt Bt . + Assume now that (τk)k≤ and (X(t), W (t))t≤τ have been defined such that a.s. • (τk)1≤k≤ is an increasing sequence of stopping times with respect to the filtration associated to X; • ∈ ≤ ≤ Xτk / Dn for 1 k ; • ≤ = for all k , there exists an integer ik such that Xik (τk) 0. + + 1 n We then define (Xt ,Bt )τ (n) (n) = (n) = { ;∃ = (n) ∈ } Lemma 5.1. Let X be a solution to the martingale problem (3), with X0 x. Let T inf t i j,Xt Dn . (n) (n),0 Then (Xt )t≤T (n) is distributed like Px . n Proof. Let x = (x1,...,xn) ∈ R \ Dn. As before I− ={j; xj ≤ 0} and I+ ={j; xj > 0}.Leti be such that xi = (n) max{xj ; j ∈ I−} when I− = ∅ and xi = min{xj ; j ∈ I} when I− = ∅.LetX be a solution of the martingale (n) = problem. This implies that for all j, Xj is a Brownian motion and for all j k, t (n) (n) X ,X = (n) (n) s. j k t 1{X (s)>0}1{X (s)>0} d (23) 0 j k 1342 Y. Le Jan and O. Raimond Let (B0,1,...,B0,n,B0) be n + 1 independent Brownian motions, and independent of X(n). Define (B1,...,Bn) by t t j (n) 0,j B = (n) X (s) + (n) B . t 1{X (s)<0} d j 1{X (s)>0} d s 0 j 0 j Note that (B1,...,Bn) are n independent Brownian motions. Set = ;∃ = (n) = τ1 inf t>0 j i, Xj (t) 0 . ≤ = 0 = ∅ Define for t τ1, Bt Bt when I+ and t t (n) 0 B = (n) X (s) + (n) B t 1{X (s)<0} d k 1{X (s)>0} d s 0 i 0 i when I+ = ∅ and where k ∈ I+ (one can choose for example k such that xk = max{xj ; j ∈ I+}). Define also for t ≤ τ1 t t + i B = (n) B + (n) B . t 1{X (s)<0} d s 1{X (s)>0} d s 0 i 0 i j Note that for t ≤ τ1, B,B t = 0 for all j and that (n) = + j ∈ ∪{ } Xj (t) xj Bt for j I− i , (n) = + + ∈ \{ } Xj (t) xj Bt for j I+ i . + Assume now that (τk)k≤ and (Bt ,Bt )t≤τ have been defined such that a.s. (n) • (τk)1≤k≤ is an increasing sequence of stopping times with respect to the filtration associated to X ; • (n) ∈ ≤ ≤ Xτk / n for 1 k ; • for all 1 ≤ k ≤ , there exists an integer i such that X(n)(τ ) = 0. k ik k + + (n) (n) = We then define (Bt ,Bt )t∈]τ ,τ +1] as is defined (Bt ,Bt )0 5.3. Proof of Theorem 2.11 + Denote by P(n) the family of consistent Feller semigroups associated to K . To this family of semigroups, we associate a unique consistent family of coalescing n-point motions, P(n),c (see Theorem 4.1 in [12]), with the property that the law of this n-point motion before hitting n is described by Lemma 5.1. These semigroups are Fellerian and since + the two-point motion hits 2 ={x = y}, the family of semigroups are associated to a coalescing SFM ϕ . Proof of (i): follow the proof of Proposition 4.8. Proof of (ii): It is a consequence of Theorem 4.2 in [12]. 5.4. Proof of Theorem 2.10 The existence (and uniqueness) of a Wiener solution K+ was proved in Section 5.1. By construction, (iii) holds. The + SFK K is diffusive since the two-point motion clearly leaves 2 (it behaves as a Brownian motion in D ={x<0 or y<0}). So (ii) is proved. To finish the proof of (i), it remains to show that K+ is the unique solution. Let K be another solution. The flow obtained by filtering K with respect to W + is a Wiener solution. Since there exists at most Three examples of Brownian flows on R 1343 + + one Wiener solution, this flow is distributed like K . For simplicity we also denote this flow K . Denote by P(2) + + + + (resp. P(2, )) the semigroup of the two-point of K (resp. of K ). Note that if P(2) = P(2, ), then K = K . Indeed, − + 2 = (2) ⊗ − + E Ks,tf(x) Ks,tf(x) Pt−s(f f )(x, x) 2E Ks,tf(x)Ks,tf(x) + (2,+) ⊗ Pt−s (f f )(x, x) = (2) ⊗ − + + Pt−s(f f )(x, x) 2E Ks,tf(x)Ks,tf(x) + (2,+) ⊗ Pt−s (f f )(x, x) = (2) ⊗ − (2,+) ⊗ Pt−s(f f )(x, x) Pt−s (f f )(x, x) = 0. Denote Dr = (x, y) ∈ R2; x ≤ 0ory ≤ 0 , + D = (x, y) ∈ R2; x ≥ 0 and y ≥ 0 , B = (x, y) ∈ R2; x = 0 and y ≥ 0 ∪ (x, y) ∈ R2; y = 0 and x ≥ 0 so that B is the boundary of Dr and D+, and R2 = Dr ∪ B ∪ D+. Let Z = (X, Y ) be a solution to the martingale problem: t 1 t ∂2 − − + f(Zt ) f (Z s) ds 1{Zs ∈D } f(Zs) ds (24) 0 2 0 ∂x ∂y ∈ 2 R2 is a martingale for all f C0 ( ).Forε>0, let + Bε = z ∈ D ; d(z,B) = ε + with d the Euclidean distance on R2 (note that when z = (x, y) ∈ D , d(z,B) = x ∧ y). r t + t r + = { ∈ r } = + Define At 0 1 Zs D ds and At 0 1{Zs ∈D } ds, the amount of time Z spends in D and D up to time t. r + r + Denote by κt and by κt the inverses of At and of At : r = ≥ ; r κt inf s 0 As >t , + = ≥ ; + κt inf s 0 As >t . Denote by Zr and by Z+ the processes in Dr and D+ defined by r + Z = Z r and Z = Z + . t κt t κt ε ε ε = ≥ Define the sequences of stopping times σk and τk by σ0 0 and for k 0 ε ε τ = inf t ≥ σ ; Zt ∈ B , k k ε = ≥ ε; ∈ ε σk+1 inf t τk Zt B . ≥ [ ε ε] ∈ + − It is easy to see that for k 1, in the time interval σk ,τk , Zt D , Yt Xt remains constant and that d(Z,B) is + = ∧ = + a Brownian motion stopped when it hits 0 (note that d(Z,B) X Y ). Thus, if Rt d(Zt ,B), then (Rt )t≤A∞ is a Brownian motion instantaneously reflected at 0 (since Rt − R0 = limε→0 ≥ (d(Z ε∧ + ,B)− d(Z ε∧ + ,B))). k 0 σk κt τk κt 1344 Y. Le Jan and O. Raimond + + Denote now by Lt the local time at 0 of Rt . Then, for any random times T , L + is the limit in probability as AT → × ε ε 0ofε NT where ε = ; ε NT max k σk r = + Denote also Lt L + . A r κt ∈ 2 R2 ∈ 2 R2 Lemma 5.2. For all f C0 ( ) (resp. f C ( )), we have t t r − 1 r − + r r f Zt f Zs ds (∂x ∂y)f Zs dLs (25) 2 0 0 is a martingale (resp. a local martingale). ∈ 2 R2 ∈ + ∈ + Proof. Take f C0 ( ) and assume Z0 D (we leave to the reader the case Z0 / D ). Then r r f Z − f Z = f(Z ε∧ r ) − f(Z ε∧ r ) (26) t 0 τk κt σk κt k≥1 + f(Z ε ∧ r ) − f(Z ε∧ r ) . (27) σk+1 κt τk κt k≥0 r ε r ε Note that for k ≥ 1 and κ >σ (which implies κ >τ ), Z ε∧ r − Z ε∧ r =−εv, with v = (1, 1). Using Taylor t k t k τk κt σk κt expansion, we have that the first term (26) is equal to ε N r κt 2 ε −ε (∂x + ∂y)f (Zσ ε ) + O ε × N r . k κt k=1 2 ε ε N r 2 ε κt 2 ε More precisely, O(ε × N r ) ≤ f 2,∞. This implies that O(ε × N r ) converges in probability towards 0 as κt 2 κt → × ε r = + ε 0 (recall ε N r converges towards Lt L + ). Thus the first term converges towards κt A r κt t − + r r (∂x ∂y)f Zs dLs 0 (it is obvious if (∂x + ∂y)f (z) = 1 when z ∈ B). Indeed: we claim that for H a bounded continuous process, + Nε A ε t H ε converges in probability towards t H dL . This holds for H = H 1] ]. Every bounded contin- k=1 σk 0 s s i ti ti−1,ti n = − − − uous process can be approached by a sequence of processes H i Hi2 n 1](i−1)2 n,i2 n]. Thus we prove the claim by density. Then we apply this result by taking Hs = (∂x + ∂y)f (Zs) to prove the convergence of the first term (26). Denote by Mf the martingale (24). Then the second term (27) is equal to f f M ε∧ r − M ε ∧ r (28) σk κt τk−1 κt k≥1 ε∧ r σ κt + 1 k f (Z s)1{Zs ∈D} ds (29) 2 ε ∧ r k≥1 τk−1 κt σ ε∧κr 2 k t ∂ + + f(Zs)1{Zs ∈D } ds. (30) ε ∧ r ∂x ∂y k≥1 τk−1 κt Three examples of Brownian flows on R 1345 Note that t = 1 r (29) f Zs ds. 2 0 + ={ ∈ +; ≤ } Note that for a constant C depending only on f , and denoting Dε z D d(z,B) ε , κr t (30) ≤ C 1 + ds {Zs ∈Dε } 0 which converges a.s. towards 0 as ε → 0. ε ∞ Denote by Mt the term (28). It is a martingale for all ε>0 and one can check that there exists a constant C< depending only on f such that d f 2 2 M = 1{ ∈ r } ∇f(Z ) + 1{ ∈ +} (∂ + ∂ )f (Z ) dt t Zt D t Zt D x y t ≤ C. Moreover, for ε <ε, ε − ε = ε − ε M M t M t M t ε ∧ r ε ∧ r σ + κt σ + κt = k 1 d f − k 1 d f M M . ε r t ε r t τ ∧κ dt τ ∧κ dt k≥0 k t k ≥0 k t { ≥ ; ∈ r ∪ +}⊃ [ ε ε ]⊃ [ ε ε ]⊃{ ≥ ; ∈ r } Since s 0 Zs D Dε k≥0 τk ,σk+1 k ≥0 τk ,σk +1 s 0 Zs D , we get that r κt ε ε M − M ≤ C 1 + ds. t {Zs ∈Dε } 0 Since this converges a.s. towards 0 as ε → 0, uniformly in ε <ε,wehavethatMε is a Cauchy sequence in the space of square integrable martingales. Thus Mε converges towards a martingale M.Thisprovesthelemma. This lemma implies that Zr is a Brownian motion with oblique reflection in the wedge Dr (see [21], it corresponds t r = π = = π = { r = } = to the case ξ 3 /2, θ1 θ2 /4 and α 1/3): 0 1 Zs 0 ds 0 and Z is a solution to the sub-martingale problem: t r − 1 r f Zt f Zs ds (31) 2 0 ∈ 2 + ≥ ∈ is a sub-martingale for all f constant in the neighborhood of 0 and f C0 (D) such that (∂x ∂y)f (z) 0forz D. r In [21], it is proved that for all initial value Z0, there is a unique solution to this sub-martingale problem. Thus the r r law of Z is uniquely determined by Z0. = r + r Applying also the lemma to the function f(x,y) x, we see that Xt Lt is a local martingale (it is actually a true martingale). This implies that Xr is a semimartingale and that in the Doob–Meyer decomposition of Xr , −Lr is its compensator. Thus Lr can be recovered from Xr . Note that this gives a proof that Zr is a semimartingale (see also [22]). r = + + =∞ + =∞ { } r =∞ Note that L r L + . Thus, if A∞ then L + (since 0 is recurrent for R) and LAr . Therefore At At A∞ ∞ + r r r r + A∞ =∞implies that A∞. On the converse, if A∞ =∞, then L∞ =∞(since B is recurrent for Z ) and L + =∞. A∞ r + r + r + Therefore A∞ =∞implies that A∞ =∞. Since A∞ + A∞ =∞,wemusthaveA∞ = A∞ =∞. It is easy to see that the processes Zr and R are independent. We have that r Zt = Z r + R + v At At 1346 Y. Le Jan and O. Raimond with v = (1, 1). = r = + r + Set Lt L r ( L + ). Define T , T and T by At At T = inf{t ≥ 0; Lt > }, r = ≥ ; r T inf t 0 Lt > , + = ≥ ; + T inf t 0 Lt > . = r + + = r + + r = { ; }= { ; r + Then T AT AT T T . Thus the process Lt is σ(Z ,R)-measurable since Lt inf T >t inf T + + + T >t}.Now,Ar = T r and A = T . Since t Lt t Lt r Zt = Z r + R + v, At At + we see that Z is σ(Zr ,R)-measurable. Thus, the law of Z is uniquely determined. This proves that P(2) = P(2, ). References [1] R. Arratia. Brownian motion on the line. Ph.D. dissertation, Univ. Wisconsin, Madison, 1979. [2] M. Benabdallah, S. Bouhadou and Y. Ouknine. On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations with jumps. Preprint, 2011. Available at arXiv:1108.4016. [3] K. Burdzy, W. Kang and K. Ramanan. The Skorokhod problem in a time-dependent interval. Stochastic Process. Appl. 119 (2009) 428–452. MR2493998 [4] K. Burdzy and H. Kaspi. Lenses in skew Brownian flow. Ann. Probab. 32 (2004) 3085–3115. MR2094439 [5] L. R. G. Fontes, M. Isopi, C. M. Newman and K. Ravishankar. The Brownian web. Proc. Natl. Acad. Sci. USA 99 (2002) 15888–15893 (electronic). MR1944976 [6] L. R. G. Fontes, M. Isopi, C. M. Newman and K. Ravishankar. The Brownian web: Characterization and convergence. Ann. Probab. 32 (2004) 2857–2883. MR2094432 [7] H. Hajri. Discrete approximations to solution flows of Tanaka’s SDE related to Walsh Brownian motion. In Séminaire de Probabilités XLIV 167–190. Lecture Notes in Math. 2046. Springer, Heidelberg, 2012. MR2953347 [8] H. Hajri. Stochastic flows related to Walsh Brownian motion. Electron. J. Probab. 16 (2011) 1563–1599 (electronic). MR2835247 [9] W. Kang and K. Ramanan. A Dirichlet process characterization of a class of reflected diffusions. Ann. Probab. 38 (2010) 1062–1105. MR2674994 [10] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Graduate Texts in Mathematics 113. Springerg, New York, 1991. MR1121940 [11] Y. Le Jan and O. Raimond. Integration of Brownian vector fields. Ann. Probab. 30 (2002) 826–873. MR1905858 [12] Y. Le Jan and O. Raimond. Flows, coalescence and noise. Ann. Probab. 32 (2004) 1247–1315. MR2060298 [13] Y. Le Jan and O. Raimond. Stochastic flows on the circle. In Probability and Partial Differential Equations in Modern Applied Mathematics 151–162. IMA Vol. Math. Appl. 140. Springer, New York, 2005. MR2202038 [14] Y. Le Jan and O. Raimond. Flows associated to Tanaka’s SDE. ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 21–34. MR2235172 [15] B. Micaux. Flots stochastiques d’opérateurs dirigés par des bruits gaussiens et poissonniens. Ph.D. dissertation, Univ. Paris-Sud, 2007. [16] R. Mikulevicius and B. L. Rozovskii. Stochastic Navier–Stokes equations for turbulent flows. SIAM J. Math. Anal. 35 (2004) 1250–1310. MR2050201 [17] R. Mikulevicius and B. L. Rozovskii. Global L2-solutions of stochastic Navier–Stokes equations. Ann. Probab. 33 (2005) 137–176. MR2118862 [18] V. Prokaj. The solution of the perturbed Tanaka equation is pathwise unique. Preprint, 2011. Available at arXiv:1104.0740. [19] K. Ramanan. Reflected diffusions defined via the extended Skorokhod map. Electron. J. Probab. 11 (2006) 934–992 (electronic). MR2261058 [20] B. Tsirelson. Nonclassical stochastic flows and continuous products. Probab. Surv. 1 (2004) 173–298 (electronic). MR2068474 [21] S. R. S. Varadhan and R. J. Williams. Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 (1985) 405–443. MR0792398 [22] R. J. Williams. Reflected Brownian motion in a wedge: Semimartingale property. Z. Wahrsch. Verw. Gebiete 69 (1985) 161–176. MR0779455