Three examples of brownian flows on <span class="mathjax-formula">$\mathbb {R}$</span>

www.imstat.org/aihp DOI:10.1214/13-AIHP541

© Association des Publications de l’Institut Henri Poincaré, 2014

Three examples of Brownian flows on R

Yves Le Jan

and Olivier Raimond

aLaboratoire Modélisation stochastique et statistique, Université Paris-Sud, Bâtiment 425, 91405 Orsay Cedex, France.

E-mail:yves.lejan@math.u-psud.fr

bLaboratoire Modal’X, Université Paris Ouest Nanterre La Défense, Bâtiment G, 200 avenue de la République 92000 Nanterre, France.

E-mail:olivier.raimond@u-paris10.fr

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) onR

dXt=1_{Xt>0}W⁺(dt)+1_{Xt<0}dW⁻(dt),

whereW⁺andW⁻are two independent white noises, is a coalescing flow we will denote byϕ^±. The flowϕ^±is a Wiener solution of the SDE. Moreover,K⁺=^E[δ_ϕ±|W⁺]is the unique solution (it is also a Wiener solution) of the SDE

K_s,t⁺f (x)=f (x)+ _t

s Ks,u

1_R+f

(x)W⁺(du)+1 2

_t s

Ks,uf(x)du

fors < t,x∈Randf a twice continuously differentiable function. A third flowϕ⁺can be constructed out of then-point motions ofK⁺. This flow is coalescing and itsn-point motion is given by then-point motions ofK⁺up to the first coalescing time, with the condition that when two points meet, they stay together. We note finally thatK⁺=^E[δ_ϕ+|W⁺].

Résumé. Nous montrons que le seul flot solution de l’équation différentielle stochastique (EDS) surR

dXt=1_{Xt>0}W⁺(dt)+1_{Xt<0}dW⁻(dt),

oùW⁺etW⁻sont deux bruits blancs indépendants, est un flot coalescent que nous noterons ϕ^±. Le flotϕ^±est une solution Wiener de l’équation. De plus,K⁺=^E[δ_ϕ±|W⁺]est l’unique solution (c’est aussi une solution Wiener) de l’EDS

K_s,t⁺f (x)=f (x)+ _t

s K_s,u

1_R+f

(x)W⁺(du)+1 2

_t s

K_s,uf(x)du

pour touts < t,x∈Retf une fonction deux fois continûment mesurable. Un troisième flotϕ⁺peut être construit à partir des mouvements ànpoints deK⁺. Ce flot est coalescent et ses mouvements ànpoints sont donnés par les mouvements ànpoints de K⁺jusqu’au premier temps de coalescence, avec comme condition que lorsque deux points se rencontrent, ils restent confondus.

On remarquera finalement queK⁺=^E[δ_ϕ+|W⁺].

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

[11–13] and [20] (see also [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 noisesW⁺andW⁻. All particles on the positive half-line are driven byW⁺ 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 noiseW, 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 ofW⁺andW⁻. 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 positiveε, 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 byW⁺ on the positive half line. A solution is easily constructed by integrating outW⁻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: then-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 byW⁺. This procedure allows to define a coalescing flow of maps, which is not Wiener measurable (i.e. notσ (W⁺)-measurable).

2. Notation, definitions and results

2.1. Notation

• Forn≥1,C(R⁺: Rⁿ)(resp.C_b(R⁺: Rⁿ)) denotes the space of continuous (resp. bounded continuous) functions f:R⁺→Rⁿ.

• Forn≥1,C0(Rⁿ)is the space of continuous functionsf:Rⁿ→Rconverging to 0 at infinity. It is equipped with the normf_∞=sup_x∈Rⁿ|f (x)|.

• Forn≥1,C₀²(Rⁿ)is the space of twice continuously differentiable functionsf :Rⁿ→Rconverging to 0 at infinity as well as their derivatives. It is equipped with the normf2,∞= f_∞+

i∂_if_∞+

i,j∂_i∂_jf_∞.

• For a metric spaceM,B(M)denotes the Borelσ-field onM.

• For n≥1, M(Rⁿ) (resp. Mb(Rⁿ)) denotes the space of measurable (resp. bounded measurable) functions f:Rⁿ→R.

• We denote byF the spaceM(R). It will be equipped with theσ-field generated byf →f (x)for allx∈R.

• P(R)denotes the space of probability measures on(R,B(R)). The spaceP(R)is equipped with the topology of narrow convergence. Forf∈Mb(R)andμ∈P(R),μf orμ(f )denotes

Rfdμ=

Rf (x)μ(dx).

• 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 byKf (x)=

Rf (y)K(x,dy). For μ∈P(R),μK∈P(R)is defined by (μK)f =μ(Kf ). IfK₁andK₂are two kernels thenK₁K₂is the kernel defined by(K₁K₂)f (x)=K₁(K₂f )(x)(= f (z)K₁(x,dy)K2(y,dz)). The spaceEwill be equipped withEtheσ-field generated by the mappingsK→μK, for everyμ∈P(R).

• We denote by(resp.⁽ⁿ⁾forn≥1) the Laplacian onR(resp. onRⁿ), acting on twice differentiable functionsf onR(resp. onRⁿ) and defined byf =f(resp.⁽ⁿ⁾f=n

i=1 ∂²

∂x²_if).

2.2. Definitions:Stochastic flows andn-point motions

Definition 2.1. A measurable stochastic flow of mappings(SFM)ϕonR,defined on a probability space(Ω,A,P), is a family(ϕs,t)s<t such that

(1) For alls < t,ϕs,tis a measurable mapping from(Ω×R,A⊗B(R))to(R,B(R));

(2) For allh∈R,s < t,ϕ_s+h,t+his distributed likeϕ_s,t;

(3) For alls < t < uand allx∈R,a.s.ϕ_s,u(x)=ϕ_t,u◦ϕ_s,t(x),andϕ_s,s equals the identity;

(4) For allf∈C0(R),ands≤t,we have

(u,v)lim→(s,t )sup

x∈RE

f ◦ϕ_u,v(x)−f ◦ϕ_s,t(x)2

=0;

(5) For allf∈C₀(R),x∈R,s < t,we have

ylim→xE

f ◦ϕ_s,t(y)−f ◦ϕ_s,t(x)2

=0;

(6) For alls < t,f ∈C₀(R), lim_|x|→∞E[(f◦ϕ_s,t(x))²] =0.

Definition 2.2. A measurable stochastic flow of kernels(SFK)KonR,defined on a probability space(Ω,A,P),is a family(K_s,t)_s<t such that

(1) For alls < t,K_s,t is a measurable mapping from(Ω×R,A⊗B(R))to(P(R),B(P(R)));

(2) For allh∈R,s < t,K_s+h,t+his distributed likeK_s,t;

(3) For alls < t < uand allx∈R,a.s.Ks,u(x)=Ks,tKt,u(x),andKs,s equals the identity;

(4) For allf∈C₀(R),ands≤t,we have

(u,v)lim→(s,t )sup

x∈R E

Ku,vf (x)−Ks,tf (x)2

=0; (5) For allf∈C0(R),x∈R,s < t,we have

ylim→xE

Ks,tf (y)−Ks,tf (x)2

=0;

(6) For alls < t,f ∈C₀(R), lim_|x|→∞E[(K_s,tf (x))²] =0.

The law of a SFK (resp. of a SFM) is a probability measure on( _s<tE,

s≤tE)(resp. on( _s<tF,

s≤tF)). A SFKKwill be called a SFM whenKs,t(x)=δϕ_s,t(x)for some SFMϕ.

Definition 2.3. Let(P⁽ⁿ⁾_t , n≥1)be a family of Feller semigroups,respectively defined onRⁿ and acting onC₀(Rⁿ) as well as on bounded continuous functions.We say that this family is consistent as soon as

(1) for alln≥1,all permutationσ of{1, . . . , n}and allf∈C0(Rⁿ), P⁽ⁿ⁾_t

f^σ

= P⁽ⁿ⁾_t fσ

,

wheref^σ(x1, . . . , xn)=f (xσ₁, . . . , xσ_n);

(2) for allk≤nand allf ∈C₀(R^k),we have P⁽ⁿ⁾_t (f◦πk,n)=

P^(k)_t f

◦πk,n,

whereπ_k,n:Rⁿ→R^k is such thatπ_k,n(x)=(x₁, . . . , x_k),

such thatx_i≤y_i fori≤n.We will denote byP⁽ⁿ⁾x the law of the Markov process associated withP⁽ⁿ⁾_t and starting fromx∈Rⁿ.This Markov process will be called then-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 SFKK and consistent family of Feller semigroupsP⁽ⁿ⁾, with the semigroupP⁽ⁿ⁾defined byP⁽ⁿ⁾_t =^E[K_0,t^⊗n]). It can be viewed as a generalization of De Finetti’s Theorem (see [20]).

2.3. Definition:White noises

LetC:R×R→Rbe a covariance function onR, i.e.Cis symmetric and

i,jλiλjC(xi, xj)≥0 for all finite family (λi, xi)∈R². Assuming that the reproducing Hilbert spaceHCassociated toCis separable, there exists(ei)i∈I (with I at most countable) an orthonormal basis ofH_Csuch thatC(x, y)=

i∈Ie_i(x)e_i(y).

Definition 2.4. A white noise of covarianceC is a centered Gaussian family of real random variables(W_s,t(x), s <

t, x∈R),such that E

W_s,t(x)W_u,v(y)

=[s, t] ∩ [u, v]×C(x, y).

A standard white noise is a centerd Gaussian family of real random variables(W_s,t, s < t )such that E[Ws,tWu,v] =[s, t] ∩ [u, v].

Starting with(Wⁱ)_i∈Iindependent standard white noises, one can defineW=(W_s,t)_s<ta white noise of covariance Cby the formula

Ws,t(x)=

i∈I

W_s,tⁱ ei(x),

which is well defined inL².

AlthoughW_s,t doesn’t belong toH_C, one can recoverW_s,tⁱ out ofW byW_s,tⁱ = W_s,t, e_i. Indeed, for any giveni, e_i is the limit asn→ ∞inH_Cofe_iⁿ=

kλⁿ_kC_xⁿ

k where for alln,(λⁿ_k, x_kⁿ)is a finite family inR². Note that e_iⁿ−e^m_i ²

HC =

k,

λⁿ_kλ^mC x_kⁿ, x^m

.

DenoteW_s,t^n,i=

kλⁿ_kW_s,t(x_kⁿ). Then fornandm, E

W_s,t^n,i−W_s,t^m,i2

=(t−s)eⁿ_i −e^m_i ²

H_C.

Thus one can defineW_s,tⁱ as the limit inL²ofW_s,t^n,i. Then one easily checks thatWs,t(x)=

iW_s,tⁱ (x)ei(x)a.s.

ForKa SFK (resp.ϕa SFM orW a white noise), we denote for alls≤t byFs,t^K (resp.Fs,t^ϕ orFs,t^W) theσ-field generated by{K_u,v;s≤u≤v≤t}(resp. by{ϕ_u,v;s≤u≤v≤t}or{W_u,v;s≤u≤v≤t}). A white noiseWis said to be aF^K (resp.F^ϕ-white noise) ifFs,t^W⊂Fs,t^K for alls < t (resp.Fs,t^W⊂Fs,t^ϕ for alls < t). In all the following, all σ-fields will be completed by negligible events.

2.4. Definition:The(¹₂, C)-SDE

Letϕbe a SFM andWaF^ϕ-white noise of covarianceC. Then if for alls < tand allx∈R, we have ϕ_s,t(x)=x+

i∈I

_t

s

e_i◦ϕ_s,u(x)Wⁱ(du) (1)

then(ϕ, W )is said to solve the SDE (1). Since this SDE is determined by the covarianceC, we will more simply say that(ϕ, W )solves theC-SDE driven byW. Note that to find SFM’s solutions of theC-SDE for which the one-point motion is a Brownian motion, we will need to assume thatC(x, x)=1 for allx∈R.

In all the following, we will be interested in constructing SFM and SFK for which the one-point motion is a Brownian motion. Adding this condition, the C-SDE will be called the(¹₂, C)-SDE, since ¹₂f =¹₂f for f ∈ C²(R)is the generator of the Brownian motion onR.

The notion of solution of this SDE can be extended to stochastic flows of kernels: let K be a SFK andW =

ie_iWⁱ, aF^K-white noise of covarianceC, then(K, W )is said to solve the(¹₂, C)-SDE driven byW if K_s,tf (x)=f (x)+

i∈I

t s

K_s,u e_if

(x)Wⁱ(du)+1 2

t s

K_s,uf(x)du (2)

for allf ∈C₀²(R),s < t andx∈R. Note that whenCis continuous, then identity (2) implies thatW is aF^K-white noise (see Section 5 and Lemma 5.3 in [12]). To find kernel solutions of the (¹₂, C)-SDE, we will only need to assume thatC(x, x)≤1 for allx∈R. (This condition comes from the fact that for allt≥0,

P_tf²=^E

K_0,tf²(x)

≥E

(K_0,tf )²(x)

=^P(2)t f^⊗2f (x, x) and that

Af²(x)−A⁽²⁾f^⊗2(x, x)=

1−C(x, x) f(x)2

,

whereA=¹₂is the generator of^Pt andA⁽²⁾is the generator of^P(2)_t .) HavingCstricly 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 of the(¹₂, C)-SDE.

Taking the expectation in (2), we see that for a solution(K, W )of the(¹₂, C)-SDE, the one-point motion ofKis a standard Brownian motion. Note that ifKis a SFK of the formδϕ, withϕa SFM, then(K, W )is a solution of the (¹₂, C)-SDE if and only if(ϕ, W )solves (1), and we must haveC(x, x)=1 for allx∈R.

In the following we will make the assumption thatC(x, x)≤1 for allx∈R.

A solution(K, W )of the(¹₂, C)-SDE is called a Wiener solution if for alls < t,Fs,t^K =Fs,t^W. A SFMϕwill be called coalescing when for allx, y,T =inf{t >0;ϕ0,t(x)=ϕ0,t(y)}is finite a.s. A SFKKwill be called diffusive when it is not a SFM.

We will say the(¹₂, C)-SDE has a unique solution when all its solutions(K, W )have the same distribution.

2.5. Martingale problems related to the(¹₂, C)-SDE

Let(K, W )be a solution of the(¹₂, C)-SDE. Denote by^P(n)_t , the semigroup associated with then-point motion of K, and byP⁽ⁿ⁾x the law of then-point motion started fromx∈Rⁿ.

Proposition 2.5. LetX⁽ⁿ⁾be distributed likeP⁽ⁿ⁾x withx∈Rⁿ.Then it is a solution of the martingale problem:

f X_t⁽ⁿ⁾

− _t

0

A⁽ⁿ⁾f X⁽ⁿ⁾_s

ds (3)

is a martingale for allf∈C₀²(Rⁿ),where A⁽ⁿ⁾f (x)=1

2⁽ⁿ⁾f (x)+

i<j

C(x_i, x_j) ∂²

∂x_i∂x_jf (x).

Proof. Whenf is of the form f1⊗ · · · ⊗f_n, withf1, . . . , f_n inC₀²(R), then it is easy to verify (3) when(K, W ) solves (2). This extends to the linear space spanned by such functions, and by density, toC₀²(Rⁿ). (A proof of this fact can be derived from the observation that, on the torus, Sobolev theorem shows that trigonometric polynomials are

dense inC².)

Remark 2.6. If one can prove uniqueness for these martingale problems,then this implies there exists at most one solution to the(¹₂, C)-SDE. (Indeed,using Itô’s formula,the expectation of any product ofK_s,tf (x)andW_s,t(y)’s can be expressed in terms ofP⁽ⁿ⁾_t ’s.)

Remark 2.7. WhenCis smooth,then the martingale problem(3)is well posed.This implies that then-point motion ofK is uniquely determined and that the(¹₂, C)-SDE has a unique solution,which is a Wiener solution.If one assumes in addition thatC(x, x)=1,then the usual theory applies,and the solution is a flow of diffeomorphisms.

2.6. Statement of the main theorems

Theorem 2.8. LetC_±(x, y)=1_{x>0}1_{y>0}+1_{x<0}1_{y<0}.Then there is a unique solution(K, W )of the(¹₂, C_±)- SDE.Moreover,this solution is a Wiener solution andKis a coalescing SFM.

Denote byϕ^±the coalescing flow defined in Theorem2.8. The white noiseW of covarianceC_±can be written in the formW=W⁺1_R++W⁻1_R−, withW⁺andW⁻two independent standard white noises. Then (2) is equivalent to

ϕ_s,t^±(x)=x+ t

s

1_{ϕ±

s,u(x)>0}W⁺(du)+ t

s

1_{ϕ±

s,u(x)<0}W⁻(du). (4)

A consequence of this theorem, with Proposition4.1below, is Theorem2.9below. 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 byB⁺andB⁻,two independent Brownian motions,

dXt=1_{X_t>0}dB_t⁺+1_{X_t<0}dB_t⁻ (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 allx∈R, there exists one and only one proba- bility measureQxonC(R⁺: R³)such that underQx(dω), the canonical process(X, B⁺, B⁻)(ω)=ωsatisfies (5), withB⁺andB⁻ two independent Brownian motions, andXa Brownian motion started atx. Proposition4.1states that (5) has a unique solution. Since (4) holds, one can take(Xt, B_t⁺, B_t⁻)=(ϕ^±_0,t(x), W_0,t⁺, W_0,t⁻)for this solution.

A solution(X, B⁺, B⁻)is a strong solution ifXis measurable with respect to theσ-field generated byB⁺ and B⁻, completed by the events of probability 0. Sinceϕ^±is a Wiener solution, one can conclude.

Theorem 2.10. LetC₊(x, y)=1_{x>0}1_{y>0}.

(i) There is a unique solution(K⁺, W )solution of the(¹₂, C₊)-SDE.

(ii) The flowK⁺is diffusive and is a Wiener solution.

(iii) The flowK⁺can be obtained by filteringϕ^±with respect to the noise generated byW⁺:K_s,t⁺ =^E[δ_ϕ± s,t|W⁺].

Theorem 2.11. There exists a unique coalescing SFMϕ⁺ such that itsn-point motions coincide with the n-point motions ofK⁺before hittingn= {x∈Rⁿ; ∃i=j, xi=xj}.Moreover

(i) (ϕ⁺, W⁺)is not a solution of the(¹₂, C_±)-SDE but it satisfies t

s

1_{ϕ+

s,u(x)>0}dϕ_s,u⁺(x)= t

s

1_{ϕ+

s,u(x)>0}W⁺(du).

(ii) K⁺can be obtained by filteringϕ⁺with respect to the noise generated byW⁺:K_s,t⁺ =^E[δ_ϕ+ s,t|W⁺].

Remark 2.12. Following[12],it should be possible to prove that he linear part of the noise generated byϕ⁺is the noise generated byW⁺.

We refer to Section3.2for more precise definitions of noises, extension of noises, filtering by a subnoise, and linear part of a noise.

3. General results

In this sectionCis any covariance function onRsatisfyingC(x, x)≤1 for allx∈R.

3.1. Chaos decomposition of Wiener solutions

Proposition 3.1. Let(K, W )be a Wiener solution of the(¹₂, C)-SDE.Then for alls < t,f a bounded measurable function onRandx∈R,a.s.,

K_s,tf (x)=

n≥0

J_s,tⁿf (x) (6)

withJⁿdefined byJ_t⁰=^Pt,whereP_t is the heat semigroup onR,and forn≥0, J_s,tⁿ⁺¹f (x)=

i

_t

s

J_s,uⁿ

(P_t−uf )e_i

(x)Wⁱ(du). (7)

This implies that there exists at most one Wiener solution to the(¹₂, 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].

Let us first remark that the stochastic integral

i

_t

0K_s(e_i(P_t−sf ))(x)Wⁱ(ds)do converge inL²for all bounded measurable functionf since (using in the fourth inequality thatC(x, x)≤1 for allx∈R)

i E

_t

s

K_s,u

(P_t−uf )e_i

(x)Wⁱ(du) 2

≤

i

_t

s E

K_s,u

(P_t−uf )e_i (x)2

du

≤

i

_t

s E

K_s,u

(P_t−uf )e_i2

(x) du

≤ _t

s

P_u−s

(P_t−uf )2

i

e_i²

(x)du

≤ _t

s P_u−s

(P_t−uf )2

(x)du

which is finite (since, using that _du^d(P_u−s(P_t−uf )²)=^Pu−s((P_t−uf ))², it is equal toP_t−sf²(x)−(P_t−sf )²(x)).

Takef aC³function with compact support. Fort >0, denoteK_0,t simplyK_t. Then fort >0,n≥1 andx∈R

Ktf (x)−^Ptf (x)=

n−1

k=0

Kt (k+1)/n(P_{t (1−(k+1)/n)}f )−Kt k/n(P_{t (1−k/n)}f )

=

n−1

k=0

(Kt (k+1)/n−Kt k/n)(P_{t (1−(k+1)/n)}f )

+

n−1

k=0

Kt k/n(P_{t (1−(k+1)/n)}f−^Pt (1−k/n)f )

=

n−1

k=0

i

t (k+1)/n t k/n

K_s

e_i(P_{t (1−(k+1)/n)}f )

(x)Wⁱ(ds)

+

n−1

k=0

t (k+1)/n t k/n

K_s 1

2(P_{t (1−(k+1)/n)}f )

(x)ds

−

n−1

k=0

K_{t k/n}(P_{t (1−k/n)}f−^Pt (1−(k+1)/n)f )

since(K, W )solves the(¹₂, C)-SDE. This last expression implies that fort >0,n≥1 andx∈R,

K_tf (x)−^Ptf (x)−

i

_t

0

K_s

e_i(P_t−sf )

(x)Wⁱ(ds)= 3

k=1

B_k(n)

with

B₁(n)=

n−1

k=0

i

_{t (k+1)/n}

t k/n

K_s

e_i(P_{t (1−(k+1)/n)}f−^Pt−sf )

(x)Wⁱ(ds),

B₂(n)= −

n−1

k=0

K_{t k/n}

P_{t (1−k/n)}f −^Pt (1−(k+1)/n)f − t

2n(P_{t (1−(k+1)/n)}f )

(x),

B₃(n)=

n−1

k=0

_{t (k+1)/n}

t k/n

(K_s−K_{t k/n}) 1

2(P_{t (1−(k+1)/n)}f )

(x)ds.

The terms in the expression ofB1(n)being orthogonal,

E

B₁(n)2

=

n−1

k=0

i

_{t (k+1)/n}

t k/n

E K_s

e_i(P_{t (1−(k+1)/n)}f−^Pt−sf )2

(x) ds

≤

n−1

k=0

i

_{t (k+1)/n}

t k/n

P_s

ei(P_{t (1−(k+1)/n)}f −^Pt−sf )2

(x)ds

≤

n−1

k=0

_{t (k+1)/n}

t k/n

P_s

(P_{t (1−(k+1)/n)}f −^Pt−sf )2

(x)ds,

where we have used Jensen inequality in the second inequality and the fact that

ie²_i(x)=C(x, x)≤1 in the last inequality. This last term is less thatnt /n

0 ^Psf−f²_∞ds=O(f²_∞t²/n).

By using triangular inequality,E[(B2(n))²]^1/2is less than

n−1

k=0 E

Kt k/n

P_{t (1−k/n)}f −^Pt (1−(k+1)/n)f − t

2n(P_{t (1−(k+1)/n)}f ) 2

(x) 1/2

≤

n−1

k=0

P_{t k/n}

P_{t (1−k/n)}f−^Pt (1−(k+1)/n)f− t

2n(P_{t (1−(k+1)/n)}f ) 2

(x) 1/2

.

Using moreover that ^Pt /nf − f − t /(2n)f_∞ = O((t/n)^3/2f_∞), we get that E[(B₂(n))²]^1/2 = O(t^3/2n^−1/2f_∞).

By using again triangular inequality,^E[(B3(n))²]^1/2is less than

n−1

k=0 E

t (k+1)/n t k/n

(K_s−K_{t k/n}) 1

2(P_{t (1−(k+1)/n)}f )

(x)ds 21/2

≤

n−1

k=0

t n

t (k+1)/n t k/n

E

(K_s−K_{t k/n}) 1

2(P_{t (1−(k+1)/n)}f ) 2

(x)

ds 1/2

.

Forf a Lipschitz function and 0≤s < t, we have (with(X, Y )the two point motion ofK of lawP⁽²⁾_(x,x)andE⁽²⁾_(x,x) the expectation with respect toP⁽²⁾_(x,x))

E

(K_tf −K_sf )²(x)

=^E(2)_(x,x)

f (X_t)−f (X_s)

f (Y_t)−f (Y_s)

≤(t−s)

Lip(f )2

with Lip(f )the Lipschitz constant off. From this estimate, and since Lip(P_{t (1−(k+1)/n)}f)≤Lip(f), we deduce thatE[(B₃(n))²]^1/2=O(t^3/2Lip(f)n^−1/2). The estimates we gave forE[(B_i(n))²], fori∈ {1,2,3}, implies that for f aC³function with compact support,

K_tf (x)=^Ptf (x)+

i

_t

0

K_s

e_i(P_t−sf )

(x)Wⁱ(ds).

This implies that

K_s,tf (x)=^Pt−sf (x)+

i

_t

s

K_s,u

(P_t−uf )

(x)Wⁱ(du). (8)

Iteratingntimes relation (8), we get that K_s,tf (x)=

n k=0

J_s,t^k f (x)+R_s,tⁿ f (x),

whereR_s,tⁿ f (x)is aL²random variable whose chaos expansion is such that itsnfirst Wiener chaoses are 0. This implies that for allk,J_s,t^k f (x)is thekth Wiener chaos ofK_s,tf (x). Thus, forf aC³function with compact support, the Wiener chaos expansion ofKs,tf (x)is given by (6). This extends to all bounded measurable functions.

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- preservingL²-continuous transformations ofΩand a familyFs,t,≤s≤t≤ ∞of sub-σ-fields ofAsuch that:

(a) T_hsendsFs,tontoFs+h,t+hfor allh∈Rands≤t, (b) Fs,tandFt,uare independent for alls≤t≤u, (c) Fs,t∧Ft,u=Fs,ufor alls≤t≤u.

Moreover,we will assume that,for alls≤t,Fs,tcontains all^P-negligible sets ofF−∞,∞,denotedF.

A subnoiseN¯ ofN is a noise(Ω,A,P, (T_h),F¯s,t), withF¯s,t⊂Fs,t. Note that a subnoise is characterized by a σ-field invariant byT_hfor allh∈R,F¯= ¯F_−∞,∞. In this case, we will say thatN¯ is the noise generated byF¯.

A linear representation ofNis a family of real random variablesX=(X_s,t;s≤t )such that:

(a) X_s,t◦T_h=X_s+h,t+hfor alls≤tandh∈R, (b) Xs,t isFs,t-measurable for alls≤t, (c) X_r,s+X_s,t=X_r,ta.s., for allr≤s≤t.

DefineF^linbe theσ-field generated by the random variablesX_s,twhereXis a linear representation ofN ands≤t.

DefineN^linto be the subnoise ofN generated byF^lin. This noise is called the linear part of the noiseN. LetP⁰ be the law of a SFK, it is a law on(Ω⁰,A⁰)=( _s<tE,

s<tE), and letK be the canonical SFK of law^P0. Forh∈R, defineT_h⁰:Ω→Ω byT_h⁰(ω⁰)s,t =ω⁰_s+h,t+h. For s < t,Fs,t⁰ is the σ-fieldFs,t^K completed by P⁰-negligible sets ofA⁰. This defines a noiseN⁰called the noise of the SFKK.

LetN¯ be a subnoise ofN⁰. In Section 3.2 in [12], a SFKK¯ is defined as the filtering ofKwith respect toN¯: for s < t,f∈C₀(R)andx∈R,K¯ is such that

K¯s,tf (x)=^E

Ks,tf (x)| ¯Fs,t

=^E

Ks,tf (x)| ¯F .

Suppose now P0 is the law of a SFK that solves the (¹₂, C)-SDE. Writing C in the form C(x, y) =

i∈Ie_i(x)e_i(y), there is aF^K-white noiseW=

i∈IWⁱe_i such that:

Ks,tf (x)=f (x)+

i∈I

_t

s

Ks,u

eif

(x)Wⁱ(du)+1 2

_t

s

Ks,uf(x)du (9)

(i.e. the(¹₂, C)-SDE is transported on the canonical space).

Proposition 3.3. LetJ ⊂I andN¯ the noise generated by{W^j;j ∈J},andW¯ =

j∈JW^je_j.LetK¯ be the SFK obtained by filteringKwith respect toN¯.Then(9)is satisfied withK replaced byK¯ andW byW¯ (orI byJ).If σ (W )¯ ⊂σ (K),¯ thenK¯ solves the(¹₂,C)-SDE with¯ C(x, y)¯ =

j∈Je_j(x)e_j(y),and driven byW¯. Proof. This proposition reduces to prove that

K¯_s,tf (x)=f (x)+

j∈J

_t

s

K¯_s,u e_jf

(x)W^j(du)+1 2

_t

s

K¯_s,uf(x)du

which easily follows by taking the conditional expectation with respect toσ (W^j;j ∈J )in equation (9).