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El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.19(2014), no. 12, 1–20.

ISSN:1083-6489 DOI:10.1214/EJP.v19-2773

Stochastic flows on metric graphs

Hatem Hajri

Olivier Raimond

Abstract

We study a simple stochastic differential equation (SDE) driven by one Brownian motion on a general oriented metric graph whose solutions are stochastic flows of kernels. Under some conditions, we describe the laws of all solutions. This work is a natural continuation of [17, 8, 10] where some particular metric graphs were considered.

Keywords: Skew Brownian motion; Stochastic flows of mappings; stochastic flows of kernels;

metric graphs.

AMS MSC 2010:Primary 60H25; Secondary 60J60.

Submitted to EJP on May 2, 2013, final version accepted on January 10, 2014.

1 Introduction

A metric graph is seen as a metric space with branching points. In recent years, dif- fusion processes on metric graphs are more and more studied [7, 12, 13, 14, 15]. They arise in many physical situations such as electrical networks, nerve impulsion propaga- tion [5, 18]. They also occur in limiting theorems for processes evolving in narrow tubes [4]. Diffusion processes on graphs are defined in terms of their infinitesimal operators in [6]. Such processes can be described as mixtures of motions “along an edge” and

“around a vertex”. Typical examples of such processes are Walsh’s Brownian motions introduced in [20] and defined on a finite number of half lines which are glued together at a unique end point. These processes have acquired a particular interest since it was proved by Tsirelson [19] that they cannot be strong solutions to any SDE driven by a standard Brownian motion, although they satisfy a martingale representation property with respect to some Brownian motion [1]. In view of this, it is natural to investigate SDEs on graphs driven by one Brownian motion to be as simple as possible. This study has been initiated by Freidlin and Sheu in [6] where any Walsh’s Brownian motion X has been shown to satisfy the equation

df(Xt) =f0(Xt)dWt+1

2f00(Xt)dt

Support: National Research Fund, Luxembourg, and Marie Curie Actions of the EC (FP7-COFUND).

Université du Luxembourg, Luxembourg.

E-mail:hatem.hajri@uni.lu

Laboratoire MODAL’X, Université Paris Ouest Nanterre La Défense, France.

E-mail:oraimond@u-paris10.fr

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whereW is the Brownian motion given by the martingale part of |X|, f runs over an appropriate domain of functions with an appropriate definition of its derivatives. Our subject in this paper is to investigate the following extension on a general oriented metric graph :

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

s

Ks,uf0(x)dWu+1 2

Z t s

Ks,uf00(x)du

whereK is a stochastic flow of kernels as defined in [16], W is a real white noise, f runs over an appropriate domain,f0andf00are defined according to an arbitrary choice of coordinates on each edge. WhenGis a star graph, this equation has been studied in [8] and whenGconsists of only two edges and two vertices the same equation has been considered in [10]. In this paper, we extend these two studies (as well as [17] where the associated graph is simply the real line) and classify the solutions on any oriented metric graph.

The content of this paper is as follows.

In Section 2, we introduce notations for any metric graphGand then define a SDE (E) driven by a white noise W, with solutions of this SDE being stochastic flows of kernels onG(Definition 2.3). Thereafter, our main result is stated. Along an edge the motion of any solution only depends onW and the orientations of the edges. The set of vertices ofGwill be denoted byV. Around a vertexv∈V, the motion depends on a flowKˆv on a star graph (associated tov) as constructed in [8].

In Section 3, starting from ( ˆKv)v∈V respective solutions to a SDE on a star graph associated to a vertex v, we construct a stochastic flow of kernels K solution of (E) under the following additional (but natural) assumption : the family ∨v∈V Fs,tKˆv; s≤t is independent on disjoint time intervals (hereFs,tKˆv is the sigma-field generated by the increments ofKˆvbetweensandt).

In Section 4, starting fromK, we recover the flows ( ˆKv)v∈V. Actually, in Sections 3 and 4, we prove more general results : the SDEs may be driven by different white noises on different edges ofG(see [9] for other applications of these results).

The main results about flows on star graphs obtained in [8] are reviewed in Section 5. Thus, as soon as the flows( ˆKv)v∈V can be defined jointly, we have a general construc- tion of a solutionK of(E). In Section 6, we consider two verticesv1andv2and under some condition only depending on the “geometry” of the star graphs associated tov1

andv2, we show that independence on disjoint time intervals of Fs,tKˆv1 ∨ Fs,tKˆv2, s≤t is equivalent to :Kˆv1 andKˆv2 are independent givenW.

Section 7 is an appendix devoted to the skew Brownian flow constructed by Burdzy and Kaspi in [3]. We will explain how this flow simplifies our construction on graphs such that any vertex has at most two adjacent edges.

Section 8 is an appendix complement to Section 5. Therein, we review the construc- tion of the flowsKˆv constructed in [8] with notations in accordance with the content of our paper.

2 Definitions and main results

2.1 Oriented metric graphs

LetGbe a metric graph as defined in [13, Section 2.1] in the sense that there exists a finite or countable setV, the set of vertices, and a partition{Ei; i∈I}ofG\V with Ia finite or countable set (i.e. G\V =∪i∈IEi and fori6=j,Ei∩Ej =∅) such that for alli∈I,Ei is isometric to an interval(0, Li), withLi≤+∞. We callEian edge,Liits

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v

Figure 1: Example of an oriented metric graph.

length and denote by{Ei, i∈I}the set of all edges onG. For any two pointsx, y∈Ei, we denote byd(x, y)the distance betweenxandyinduced by the isometry. We assume thatGis arc connected and for anyx, y∈G, we denote byd(x, y)the natural metric on Gobtained as the minimal length of all paths fromxtoy.

To each edge Ei, we associate an isometry ei : Ji → E¯i, with Ji = [0, Li] when Li<∞andJi= [0,∞)orJi= (−∞,0]whenLi=∞. Note thatei(t)∈Eifor alltin the interior ofJi,ei(0)∈V and whenLi <∞,ei(Li)∈V. The mappingeiwill be called the orientation of the edgeEi and the familyE ={ei; i∈ I}defines the orientation ofG. WhenLi <∞, set{gi, di} ={ei(0), ei(Li)}. When Ji = [0,∞), set{gi, di} = {ei(0),∞}

and whenJi = (−∞,0]set{gi, di} ={∞, ei(0)}. For allv∈V, setIv+={i∈I; gi=v}, Iv = {i ∈ I; di = v} and Iv = Iv+∪Iv. Let nv, n+v and nv denote respectively the numbers of elements inIv,Iv+andIv. Thennv =n+v +nv.

We will always assume that

• nv<∞for allv∈V (i.e. Iv is a finite set);

• infiLi=L >0.

A graph with only one vertex and such thatLi=∞for alli∈I will be called a star graph. It will also be convenient to imbed any star graph in the complex planeC. Its unique vertex will be denoted by0.

For eachv∈V, defineGv ={v} ∪ ∪i∈IvEi andGLv ={x∈G; d(x, v)< L}, which is then a subset ofGv. Note thatGv∩V ={v}. For eachv∈V, there exists a star graph Gˆvand a mappingiv:Gv →Gˆvsuch thativ:Gv→iv(Gv)is an isometry. This implies in particular thativ(v) = 0and thatGˆLv ={x∈Gˆv; d(0, v)< L}=iv(GLv). For eachi∈Iv, defineeˆvi =iv◦ei. Note thatGˆvcan be written in the form{0}∪∪i∈Iviv, withiv(Ei)⊂Eˆiv and where ( ˆEiv)i∈Iv is the set of edges of Gˆv. The mappingeˆvi can be extended to an isometry(−∞,0]→ {0} ∪Eˆiv wheni∈Ivand to an isometry[0,+∞)→ {0} ∪Eˆivwhen i∈Iv+.

For x ∈ Gv and f : Gv → R, setv := iv(x) and let fˆv : ˆGv → Rbe the mapping defined byfˆ= 0 oniv(Gv)c and fˆv = f ◦i−1v on iv(Gv), so that fˆv(ˆxv) = f(x)for all x∈Gv.

We will also denote byB(G)the set of Borel sets ofGand byP(G)the set of Borel probability measures onG. Recall that a kernel onGis a measurable mappingk:G→ P(G). Forx ∈ Gand A ∈ B(G), k(x, A)denotesk(x)(A) and the probability measure k(x)will sometimes be denoted byk(x, dy). Forf a bounded measurable mapping on

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0

Figure 2: The star graphGˆvassociated tovin Figure 1.

G,kf(x)denotesR

f(y)k(x, dy). The set of all continuous functions onGwhich vanish at infinity will be denoted byC0(G).

2.2 SDE onG

Let G be an oriented metric graph. To each v ∈ V and i ∈ Iv, we associate a transmission parameterαiv>0such thatP

i∈Ivαiv = 1and setα= (αiv; v∈V, i∈Iv). DefineDαGthe set of all continuous functionsf :G→Rsuch that for alli∈I,f◦ei is C2on the interior of Ji with bounded first and second derivatives both extendable by continuity toJiand such that for allv∈V

X

i∈Iv+

αiv lim

r→0+(f◦ei)0(r) = X

i∈Iv,Li<∞

αiv lim

r→Li(f ◦ei)0(r)

+ X

i∈Iv,Li=∞

αiv lim

r→0−(f ◦ei)0(r).

Since αwill be fixed, DGα will simply be denoted by D. When Gˆv is a star graph as defined before, to the half lineEˆiv, we associate the parameterαiv. Setαv= (αiv; i∈Iv) andDˆv=DGαˆvv. Forf ∈ Dandx=ei(r)∈G\V, setf0(x) = (f◦ei)0(r),f00(x) = (f◦ei)00(r) and take the conventionf0(v) =f00(v) = 0for allv∈V.

Definition 2.1. A stochastic flow of kernels (SFK) K on G, defined on a probability space(Ω,A,P), is a family(Ks,t)s≤tsuch that

1. For alls≤t,Ks,tis a measurable mapping from(G×Ω,B(G)⊗A)to(P(G),B(P(G))); 2. For allh∈R,s≤t,Ks+h,t+his distributed likeKs,t;

3. For alls1≤t1≤ · · · ≤sn ≤tn, the family{Ksi,ti,1≤i≤n}is independent;

4. For all s ≤ t ≤ uand all x∈ G, a.s. Ks,u(x) = Ks,tKt,u(x), andKs,s equals the identity;

5. For allf ∈C0(G), ands≤t, we have lim

(u,v)→(s,t)sup

x∈GE[(Ku,vf(x)−Ks,tf(x))2] = 0;

6. For allf ∈C0(G),x∈G,s≤t, we have

y→xlimE[(Ks,tf(y)−Ks,tf(x))2] = 0;

7. For alls≤t,f ∈C0(G),lim|x|→∞E[(Ks,tf(x))2] = 0.

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We say thatϕis a stochastic flow of mappings (SFM) on GifKs,t(x) = δϕs,t(x) is a SFK onG. Given two SFK’sK1 andK2 onG, we say thatK1is a modification ofK2 if for alls≤t,x∈G, a.s.Ks,t1 (x) =Ks,t2 (x).

For a family of random variables Z = (Zs,t)s≤t, set for alls≤ t, Fs,tZ =σ(Zu,v, s≤ u≤v≤t).

Definition 2.2. (Real white noise) A family(Ws,t)s≤tis called a real white noise if there exists a Brownian motion on the real line(Wt)t∈R, that is(Wt)t≥0and(W−t)t≥0are two independent standard Brownian motions such that for all s ≤ t, Ws,t = Wt−Ws (in particular, whent≥0,Wt=W0,tandW−t=−W−t,0).

Our main interest in this paper is the following SDE, that extends Tanaka’s SDE to metric graphs.

Definition 2.3. (Equation(EαG)) On a probability space(Ω,A,P), letW be a real white noise andKbe a stochastic flow of kernels onG. We say that(K, W)solves(EαG)if for alls≤t,f ∈ Dandx∈G, a.s.

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

s

Ks,uf0(x)W(du) +1 2

Z t s

Ks,uf00(x)du.

Whenϕis a SFM andK=δϕis a solution of(E), we simply say that(ϕ, W)solves(EαG). SinceGand αwill be fixed from now on, we will denote Equation(EαG)simply by (E), and we will also denote(EαGˆvv)simply by( ˆEv). A complete classification of solutions to( ˆEv)has been given in [8].

A family ofσ-fields(Fs,t; s ≤t)will be said independent on disjoint time intervals (abbreviated : i.d.i) as soon as for all (si, ti)1≤i≤n with si ≤ ti ≤ si+1, the σ-fields (Fsi,ti)1≤i≤n are independent. Note that for K a SFK, since the increments ofK are independent, then(Fs,tK; s≤t)is i.d.i.

Our main result is the following

Theorem 2.4. (i)Let W be a real white noise and let( ˆKv)v∈V be a family of SFK’s respectively onGˆv. Assume that, for eachv∈V,( ˆKv, W)is a solution of( ˆEv)and that

s,t:=∨v∈VFs,tKˆv; s≤t

is independent on disjoint time intervals. Then, there exists a unique (up to modification) SFKKonGsuch that

• Fs,tK ⊂Fˆs,tfor alls≤t,

• (K, W)is a solution to(E)and

• for alls∈Randx∈Gv, setting

ρx,vs = inf{u≥s: Ks,u(x, Gv)<1}, (2.1) then for allt > s, a.s. on the event{t < ρx,vs },

iv∗Ks,t(x) = ˆKs,tv (ˆxv). (2.2) (ii)Let(K, W)be a solution of(E). Then for eachv∈V, there exists a unique (up to modification) SFKKˆv onGˆvsuch that

• for alls≤t,Fˆs,t:=∨v∈VFs,tKˆv ⊂ Fs,tK,

• ( ˆKv, W)is a solution of( ˆEv)for eachv∈V

and such that ifρx,vs is defined by (2.1), then for alls < t inRand x∈Gv, a.s. on the event{t < ρx,vs }, (2.2) holds.

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Note that (2.2) can be rewritten : for all bounded measurable functionsf onG, and allx∈Gv

Ks,tf(x) = ˆKs,tv(ˆxv).

Theorem 2.4 reduces the construction of solutions to(E)to the construction of solu- tions to( ˆEv). Since for allv∈V, all solutions to( ˆEv)are described in [8], to complete the construction of all solutions to(E), one has to be able to construct them jointly. The classification is therefore complete if one wants to construct flows K solutions of(E) such that the flowsKˆv associated toKare independent givenW.

Since for allv∈V, there is a uniqueσ(W)-measurable flow solving( ˆEv), Theorem 2.4 implies that there is a uniqueσ(W)-measurable flow solving(E). Notice also that in general, the condition Fˆs,t;s≤t

is i.d.i does not imply that the flowsKˆv are inde- pendent givenW. And thus, even thought for all v ∈V there exists a unique (in law) flow of mappings solution of ( ˆEv), it may be possible, assuming only that Fˆs,t;s ≤t is i.d.i, to construct different (in law) flows of mappings solving(E). However, there is a unique (in law) flow of mappings solution to(E)such that the associated flows of mappings solutions to( ˆEv)are independent givenW.

For eachv ∈V, letα+v =P

i∈I+v αivandβv = 2α+v −1. Under some condition linking βv1 andβv2, the next proposition shows that( ˆFs,t; s ≤t)is i.d.i if and only ifKˆv1 and Kˆv2are independent givenW.

Proposition 2.5. Letv1andv2be two vertices inV such thatβv2 6=βv1 and

v2−βv1| ≥2βv1βv2.

LetW be a real white noise. LetKˆv1 andKˆv2 be SFK’s respectively onGˆv1 and onGˆv2 such that( ˆKv1, W)and( ˆKv2, W)are solutions respectively to( ˆEv1)and to( ˆEv2). Then (Fs,tKˆv1 ∨ Fs,tKˆv2)s≤tis i.d.i if and only ifKˆv1 andKˆv2 are independent givenW.

This proposition has been proved in [10] in the case whereV = {v1, v2}, withGˆv1 andGˆv2 being given by the following star graphs

1/2 1/2 1/2

1/2 V1 V2

Figure 3:Gˆv1 andGˆv2.

3 Construction of a solution of (E) out of solutions of ( ˆ E

v

)

For alli∈I, letWi be a real white noise. Assume thatW := (Ws,ti ; i∈I, s≤t)is Gaussian. Let

As,t:={sup

i∈I

sup

s<u<v<t

|Wu,vi |< L}. (3.1) Assume thatlim|t−s|→0+P(Acs,t) = 0. Note that this assumption is satisfied ifWi =W for alli, or ifIis finite.

LetKˆ = ( ˆKv)v∈V be a family of SFK’s respectively onGˆvand letWv:= (Wi; i∈Iv). Assume that( ˆKv,Wv)is a solution to the following SDE : for alls≤t,fˆ∈Dˆv,xˆ∈Gˆv, a.s.

s,tv fˆ(ˆx) = ˆf(ˆx) +X

i∈Iv

Z t s

s,uv (1Eˆv i

0)(ˆx)Wi(du) +1 2

Z t s

s,uv00(ˆx)du. (3.2) Then we have the following

Lemma 3.1. For allv∈V,i∈Iv and alls≤t, we haveFs,tWi ⊂ Fs,tKˆv.

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Proof. Lety = ˆevi(r)∈Eˆvi. Following Lemma 6 [8], we prove thatKˆs,tv (y) =δeˆv

i(r+Ws,ti )

for alls≤t≤σsywhere

σsy= inf{u≥s; ˆevi(r+Ws,ui ) = 0}.

Since this holds for arbitrarily larger, the lemma is proved.

In all this section, we assume that

s,t:=∨v∈VFs,tKˆv; s≤t

is i.d.i. (3.3)

We will prove the following

Theorem 3.2. There existsKa unique (up to modification) SFK onG, such that

• Fs,tK ⊂Fˆs,tfor alls≤t,

• (K, W)is a solution to the SDE : for alls≤t,f ∈ Dandx∈G, a.s.

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

i∈I

Z t s

Ks,u(1Eif0)(x)Wi(du) +1 2

Z t s

Ks,uf00(x)du

and such that, defining fors∈R,v∈V andx∈Gv,

ρx,vs = inf{t≥s: Ks,t(x, Gv)<1}, (3.4) we have that for alls < tinRandx∈Gv, a.s. on the event{t < ρx,vs },

iv∗Ks,t(x) = ˆKs,tv (ˆxv). (3.5) Note that Theorem 3.2 implies (i) of Theorem 2.4.

3.1 Construction ofK

For alls∈Randx∈G, define

τsx= inf{t≥s; ei(r+Ws,ti )∈V}

wherei ∈ I and r ∈ Ji are such that x = ei(r). For s < t, define the kernel Ks,t0 on Gby : on the eventAcs,t, setKs,t0 (x) = δx, and on the eventAs,t, ifx =ei(r)∈ Gand v=ei(r+Ws,τi x

s), set

Ks,t0 (x) =

( δei(r+Wi

s,t), if t≤τsx i−1v ∗Kˆs,tv (ˆxv), if t > τsx (i.e. forA ∈ B(G), i−1v ∗Kˆs,tv (ˆxv)

(A) = ˆKs,tv (ˆxv, iv(A∩Gv))). Note that onAs,t∩ {t >

τsx} ∩ {v=ei(r+Ws,τi x

s)}, we have that the support ofKˆs,tv (ˆxv)is included iniv(Gv)so thatKs,t0 (x)∈ P(G). Remark also that onAs,t∩ {v=ei(r+Ws,τi x

s)}, a.s.

Ks,t0 (x) =i−1v ∗Kˆs,tv (ˆxv).

Lemma 3.3. For alls < t < uand allµ∈ P(G), a.s. onAs,u,

µKs,u0 =µKs,t0 Kt,u0 . (3.6)

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Proof. Fixs < t < uand note thatAs,u⊂As,t∩At,u a.s. We will prove the lemma only forµ=δxwhich is enough since by Fubini’s Theorem : ∀A∈ B(G)

E[|µKs,u0 (A)−µKs,t0 Kt,u0 (A)|]≤ Z

G

E[|Ks,u0 (x, A)−Ks,t0 Kt,u0 (x, A)|]µ(dx).

Letiandrbe such thatx=ei(r).

When t ≤ τsx, set Y = ei(r+Ws,ti ). If u ≤ τsx, then it is easy to see that (3.6) holds after having remarked that τtY = τsx. If t ≤ τsx < u, then Ks,t0 (x) = δY and Ks,u0 (x) =i−1v ∗Kˆs,uv (ˆxv)withv=ei(r+Ws,τi x

s). We still haveτtYsxwhich is now less thanu. WriteKs,t0 Kt,u0 (x) =Kt,u0 (Y) = i−1v0 ∗Kˆt,uv0( ˆYv)wherev0 = ei(e−1i (Y) +Wt,τi Y

t

). Sincee−1i (Y) =r+Ws,ti , we have

v0=ei(r+Ws,ti +Wt,τi Y t

) =ei(r+Ws,τi x s) =v.

SinceKˆvis a flow, we get

Ks,t0 Kt,u0 (x) =i−1v ∗Kˆt,uv ( ˆYv) =i−1v ∗Kˆs,tvt,uv (ˆxv) =i−1v ∗Kˆs,uv (ˆxv) =Ks,u0 (x).

Whent > τsx, thenKs,t0 (x) =i−1v ∗Kˆs,tv (ˆxv)andKs,u0 (x) =i−1v ∗Kˆs,uv (ˆxv)withvdefined as above. Letf be a bounded measurable function onG. ThenKs,u0 f(x) = ˆKs,uvv(ˆxv). SinceKˆvis a flow, we have

Ks,u0 f(x) = ˆKs,tvt,uvv(ˆxv).

Note that on the event As,t∩ {τsx < t}, the support of Ks,t0 (x) is included in GLv, and for ally in the support of Ks,t0 (x), the support ofKˆt,uv (ˆyv) is included inGˆLv. In other words, it holds that on the event As,t∩ {τsx < t}, for all y in the support of Ks,t0 (x), Kˆt,uvv(ˆyv) = Kt,u0 f(y) and thus that Kˆs,tvt,uvv(ˆyv) = Ks,t0 Kt,u0 f(y). This implies the lemma.

We will say that a random kernelKisFellerianwhen for alln≥1and allh∈C0(Gn), we haveE[K⊗nh]∈C0(Gn).

Lemma 3.4. For alls < t,Ks,t0 is Fellerian.

Proof. By an approximation argument (see the proof of Proposition 2.1 [16]), it is enough to prove the followingL2-continuity for K0 : for all f ∈ C0(G)and all x∈ G, limy→xE[(K0,t0 f(y)−K0,t0 f(x))2] = 0. Write

(K0,t0 f(y)−K0,t0 f(x))2= (K0,t0 f(y)−K0,t0 f(x))21A0,t+ (f(y)−f(x))21Ac0,t. Suppose thatxbelongs to an edgeEi. Using the convergence in probabilityWτiy

0

→Wτix 0

asy → x, we see thatP(K0,τ0 y 0

(y)6= K0,τ0 x

0(x))converges to0 asy → x. To conclude, it remains to prove that for v ∈ {gi, di} (i.e. v is an end point of Ei), letting Ctv = A0,t∩ {K0,τ0 y

0

(y) =K0,τ0 x

0(x) =δv}, we have

y→xlimE[(K0,t0 f(y)−K0,t0 f(x))21Cvt] = 0.

Since onCtv,K0,t0 (z) =i−1v ∗Kˆ0,tv (ˆzv)forz∈ {x, y}, our result holds.

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Lemma 3.5. Let K1 and K2 be two independent Fellerian kernels. Then K1K2 is a Fellerian kernel.

Proof. Set P(n)1 = E[K1⊗n] and P(n)2 = E[K2⊗n]. Then P(n)1 P(n)2 = E[(K1K2)⊗n]. This implies the lemma.

Define forn∈N,Dn :={k2−n; k∈Z}. Fors∈R, letsn = sup{u∈Dn; u≤s}and s+n =sn+ 2−n. For everyn≥1ands≤tdefine

Ks,tn =Ks,s0 + nKs0+

n,s+n+2−n. . . Kt0

n−2−n,tnKt0n,t,

ifs+n ≤tandKs,tn =Ks,t0 , ifs+n > t. Note that Lemma 3.4 and Lemma 3.5 imply thatKs,tn is Fellerian (since the kernelsK0

s,s+n,K0

s+n,s+n+2−n,. . . , Kt0

n−2−n,tn,Kt0n,t are independent by (3.3)).

DefineΩns,t={supisup{s<u<v<t;|v−u|≤2−n}|Wu,vi |< L}. Note that for alls≤u < v≤t such that |u−v| ≤ 2−n, we have Ωns,t ⊂ Au,v. Let Ωs,t = ∪nns,t, thenP(Ωs,t) = 1. Define now, forω ∈ Ωs,t, Ks,t(ω) = Ks,tn (ω)wheren =ns,t = inf{k; ω ∈Ωks,t}and set Ks,t(x) =δxonΩcs,t.

Lemma 3.6. For alls < tand allµ∈ P(G), a.s. we have µKs,tm =µKs,t for allm≥ns,t. Proof. Form≥ns,t, we have (denotingn=ns,t)

µKs,t=µKs,s0 + nKs0+

n,s+n+2−n. . . Kt0

n−2−n,tnKt0n,t.

wheres+n, s+n + 2−n,· · ·, tn are also inDm. Moreover, for all(u, v)∈ {(s, s+n),(s+n, s+n + 2−n),· · · ,(tn, t)}, we haveΩns,t⊂Au,v. Now applying Lemma 3.3 and an independence argument, we see thatµKs,t=µKs,tm.

Proposition 3.7. Kis a SFK.

Proof. Obviously the increments of K are independent. Fix s < t < u, then by the previous lemma and Lemma 3.3 a.s. formlarge enough (i.e. m≥max{ns,u, ns,t, nt,u}), we have

µKs,u = µKs,um =µKs,tmmKtm

m,t+mKtm+

m,t+m+2−m· · ·Kumm,u

= µKs,tm

mKtm

m,tKm

t,t+mKm

t+m,t+m+2−m· · ·Kum

m,u

= µKs,tmKt,um

= µKs,tKt,u.

This proves thatKsatisfies the flow property.

Fixk≥1,h∈C0(Gk). Let >0andn1∈Nsuch thatP(ns,t> n1)< . Then for all (x, y)∈Gk×Gk, sinceP(Ωs,t) = 1, we have

|E[Ks,t⊗kh(y)]−E[Ks,t⊗kh(x)]| ≤ X

n≤n1

E

(Ks,tn )⊗kh(y)−(Ks,tn )⊗kh(x)212 + 2khk.

Now sinceKs,tn is Fellerian for alln, we deduce that lim sup

y→x

|E[Ks,t⊗kh(y)]−E[Ks,t⊗kh(x)]| ≤2khk. Sinceis arbitrary, it holds that for alls < t,Ks,tis Fellerian.

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Lemma 3.8. For allx∈Gandf ∈C0(G),lim|t−s|→0E[(Ks,tf(x)−f(x))2] = 0.

Proof. Take x = ei(r) and let > 0. Then, there exists α > 0 such that |t−s| < α implies P(As,t) > 1−. Note that a.s. on As,t, Ks,t(x) = Ks,t0 (x). If x 6∈ V, then E[(Ks,tf(x)−f(x))21As,t]≤2kfk2P(τsx< t) +E[(f(ei(r+Ws,ti ))−f(e(r)))21t≤τx

s]. The two right hand terms clearly converge to0as|t−s|goes to0. This implies the lemma whenx6∈V. Whenx=v∈V, then a.s. onAs,t,Ks,tf(x) = ˆKs,tvv(0). We can conclude the proof now sinceKˆvis a SFK.

Lemma 3.8 together with the flow property implies that for all f ∈ C0(G)and all x∈G, (s, t)7→Ks,tf(x)is continuous as a mapping from{s < t} →L2(P). Now since for alls < tinD, the law ofKs,tonly depends on|t−s|, the continuity of this mapping implies that this also holds for alls < t. Thus, we have proved thatKis a SFK.

3.2 The SDE satisfied byK

Recall that each flowKˆvsolves equation( ˆEv)defined onGˆv. Then we have Lemma 3.9. For allx∈G,f ∈ Dand alls < t, a.s. onAs,t

Ks,t0 f(x) =f(x) +X

i∈I

Z t s

Ks,u0 (1Eif0)(x)Wi(du) +1 2

Z t s

Ks,u0 f00(x)du.

Proof. Let x= ei(r)withi ∈ Iv. Recall the notationxˆv =iv(x)∈ Gˆv. Then denoting Bs,tv =As,t∩ {τsx≤t} ∩ {ei(r+Ws,τi x

s) =v}, we have that a.s. onAs,t, Ks,t0 f(x) = (f◦ei)(r+Ws,ti )1x

s>t}+X

v∈V

s,tvv(ˆxv)1Bv

s,t. Thus a.s. onAs,t,

Ks,t0 f(x) = f(x) + 1x

s>t}

Z t s

(f◦ei)0(r+Ws,ui )Wi(du) +1 2

Z t s

(f◦ei)00(r+Ws,ui )du

+ X

v∈V

1Bvs,t

 X

j∈Iv

Z t s

s,uv 1Eˆvj( ˆfv)0

(ˆxv)Wj(du) + Z t

s

s,uv ( ˆfv)00(ˆxv)du

= f(x) + 1x s>t}

Z t s

Ks,u0 (1Eif0)(x)Wi(du) +1 2

Z t s

Ks,u0 f00(x)du

+X

v∈V

1Bvs,t

 X

j∈Iv

Z t s

Ks,u0 (1Ejf0)(x)Wj(du) + Z t

s

Ks,u0 f00(x)du

.

This implies the lemma.

Lemma 3.10. For alln∈N,x∈G,s < tand allf ∈ Da.s. onΩns,t, we have

Ks,tn f(x) = f(x) +X

i∈I

Z t s

Ks,un (1Eif0)(x)Wi(du)

+ 1 2

Z t s

Ks,un f00(x)du.

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Proof. We proceed by induction onq=Card{s, s+n, s+n+ 2−n,· · · , tn, t}. Forq= 2, this is immediate from Lemma 3.9 sinceΩns,t⊂As,t. Assume this is true forq−1and lets < t such that Card{s, s+n, s+n + 2−n,· · ·, tn, t}=q.Then a.s.

Ks,tn f(x) = Ks,tn

nKtn

n,tf(x)

= Ks,tn n f +X

i∈I

Z t tn

Ktnn,u(1Eif0)Wi(du) +1 2

Z t tn

Ktnn,uf00du

! (x)

= Ks,tn

nf(x)

+ X

i∈I

Z t tn

Ks,tnnKtnn,u(1Eif0)(x)Wi(du) +1 2

Z t tn

Ks,tn nKtnn,uf00(x)du

= f(x) +X

i

Z t s

Ks,un (1Eif0)(x)Wi(du) +1 2

Z t s

Ks,un f00(x)du, by independence of increments and using the fact thatKs,tn

n(x)is supported by a finite number of points.

Thus we have

Lemma 3.11. For allx∈G,f ∈ Dand alls < t, a.s.

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

i∈I

Z t s

Ks,u(1Eif0)(x)Wi(du) +1 2

Z t s

Ks,uf00(x)du. (3.7) Proof. Note that for alln, onΩns,t, for allu∈[s, t], a.s. Ks,u(x) = Ks,un (x). Thus a.s. on Ωns,t, (3.7) holds inL2(P)and finally a.s. (3.7) holds.

Remark : WhenWi=W for alli, then(K, W)solves the SDE (E).

Lemma 3.11 with the fact thatKis a SFK permits to prove thatKsatisfies the first two conditions of Theorem 3.2. Note that for alls≤t and allx∈G, we have that a.s.

onAs,t, Ks,t(x) =Ks,t0 (x). Thus a.s., onAs,t, (3.5) holds. Now, we want to prove that a.s. (3.5) holds on the event{t < ρx,vs }(note that a.s. As,t∩ {t > τsx} ⊂ {τsx< t < ρx,vs }).

By Lemma 3.6, a.s. for allm≥ns,tsuch thats+m≤t, Ks,t(x) =Ks,s0 +

mKs0+

m,s+m+2−m. . . Kt0

m−2−m,tmKt0m,t(x).

Clearly on{t < ρs,vs }, a.s. K0

s,s+m(x) = i−1v ∗Kˆv

s,s+m(ˆxv) and for all y in the support of K0

s,s+m(x),K0

s+m,s+m+2−m(y) =i−1v ∗Kˆv

s+m,s+m+2−m(ˆyv). Thus, on{t < ρs,vs }, a.s.

Ks,s0 + mKs0+

m,s+m+2−m(x) =i−1v ∗Kˆs,sv + m

sv+

m,s+m+2−m(ˆxv).

The same argument shows that on{t < ρs,vs }, a.s.

Ks,t(x) =i−1v ∗Kˆs,sv + m

sv+

m,s+m+2−m. . .Kˆtv

m−2−m,tmtv

m,t(ˆxv) =i−1v ∗Kˆs,tv (ˆxv).

To conclude the proof of Theorem 3.2, it remains to prove that ifK0is a SFK satisfying also the conditions of Theorem 3.2, thenK0 is a modification ofK. Since (3.5) holds for Kand K0, for all s≤tand allµ∈ P(G)a.s. onAs,t,µKs,t0 =µKs,t(=µKs,t0 ). Thus for alls≤tandx∈G, denotingn=ns,t, a.s.

Ks,t0 (x) = Ks,s0 +

n· · ·Kt0n,t(x)

= Ks,s+

n· · ·Ktn,t(x)

= Ks,t(x).

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4 Construction of solutions of ( ˆ E

v

) out of a solution of (E) .

Let W = (Wi; i∈ I)be as in the previous section. Let K be a SFK onG. Assume that(K, W)satisfies the SDE : for alls≤t,f ∈ D,x∈G, a.s.

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

i∈I

Z t s

Ks,u(1Eif0)(x)Wi(du) +1 2

Z t s

Ks,uf00(x)du.

Following [10, Lemma 3], one can prove thatFs,tWi ⊂ Fs,tK for alli∈Iands≤t. In this section, we will prove the following

Theorem 4.1. For eachv ∈ V, there exists a unique (up to modification) SFKKˆv on Gˆvsuch that

• for alls≤t,Fˆs,t:=∨v∈VFs,tKˆv ⊂ Fs,tK,

• for allv∈V,( ˆKv,Wv)is a solution to the SDE : for alls≤t,fˆ∈Dˆv,xˆ∈Gˆv, a.s.

s,tv fˆ(ˆx) = ˆf(ˆx) +X

i∈Iv

Z t s

s,uv (1Eˆvi0)(ˆx)Wi(du) +1 2

Z t s

s,uv00(ˆx)du. (4.1)

and such that defining fors∈Randx∈Gvx,vs by (3.4), we have that for allt > s, a.s.

on the event{t < ρx,vs }, (3.5) holds.

Proof. Fixv ∈V. Fors∈Randxˆ = ˆevi(r)∈Gˆv withxˆ∈iv(Gv), setx=ei(r)(and we havexˆ=iv(x)). Recall the definition ofτsx. Recall also the definition ofAs,tfrom (3.1).

Define the kernelKˆs,t0,vby :

• OnAcs,t:Kˆs,t0,v(ˆx) =δxˆ;

• OnAs,t: Letxˆ = ˆevi(r). Ifˆx= ˆevi(r)∈iv(Gv),τsx < tandei(r+Ws,τi x

s) =v, define Kˆs,t0,v(ˆx) =iv∗Ks,t(ei(r)). And otherwise, defineKˆs,t0,v(ˆx) =δˆev

i(r+Ws,ti ). Now forn≥1, set

s,tn,v= ˆK0,v

s,s+n

0,v

s+n,s+n+2−n. . .Kˆt0,v

n−2−n,tnt0,v

n,t, ifs+n ≤tandKˆs,tn,v= ˆKs,t0,v, ifs+n > t.

DefineΩns,t, Ωs,t and ns,tas in Section 3.1 and finally set Kˆs,tv = ˆKs,tn,v, wheren = ns,t

and Kˆs,tv (ˆx) = δˆx onΩcs,t. Following Sections 3.1 and 3.2, we prove that Kˆv is a SFK satisfying (4.1). Note that for alls ≤t, x∈Gv, Ks,t(x) = i−1v ∗Kˆs,t0,v(ˆxv). Since for all s≤t andxˆ∈Gˆv, a.s. onAs,t,Kˆs,tv = ˆKs,t0,v, the last statement of the theorem holds. It remains to remark the uniqueness up to modification, which can be proved in the same manner as for Theorem 3.2.

The previous theorem implies (ii) of Theorem 2.4.

5 Stochastic flows on star graphs [8].

In this section, we overview the content of [8] where equation(EαG)has been studied whenGis a star graph. Let G = {0} ∪ ∪i∈IEi be a star graph where I = {1,· · · , n}. Assume thatI+={i:gi = 0}={1,· · ·, n+}andI={i:di = 0}={n++ 1,· · ·, n}and setn=n−n+. To each edgeEi, we associateαi∈[0,1]such thatP

i∈Iαi = 1. Denote

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