E l e c t ro n ic J o f P r o b 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 2f
00(X t)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.
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 F ˆ Kv s,t; s ≤ t
is independent on disjoint time intervals (hereFKˆv
s,t 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 soluconstruc-tionK 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 F ˆ Kv1 s,t ∨ F ˆ Kv2 s,t , 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 fori 6= j,Ei∩ Ej = ∅) such that for
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 → ¯Ei, 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 n−v denote respectively the
numbers of elements inIv,Iv+andIv−. Thennv = n+v + n−v.
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 andG L
v = {x ∈ G; d(x, v) < L}, which is
then a subset ofGv. Note thatGv∩ V = {v}. For eachv ∈ V, there exists a star graph ˆ
Gvand a mappingiv: Gv → ˆGvsuch thativ: Gv→ iv(Gv)is an isometry. This implies in
particular thativ(v) = 0and thatGˆLv = {x ∈ ˆGv; d(0, v) < L} = iv(GLv). For eachi ∈ Iv,
defineeˆv
i = iv◦ei. Note thatGˆvcan be written in the form{0}∪∪i∈IvEˆ v
i, withiv(Ei) ⊂ ˆEiv
and where ( ˆEv
i)i∈Iv is the set of edges of Gˆ
v. The mappingeˆv
i can be extended to an
isometry(−∞, 0] → {0} ∪ ˆEiv wheni ∈ Iv−and to an isometry[0, +∞) → {0} ∪ ˆEivwhen i ∈ Iv+.
For x ∈ Gv and f : Gv → R, set xˆv := iv(x) and let fˆv : ˆGv → R be the mapping
defined byf = 0ˆ oniv(Gv)c and fˆv = f ◦ iv−1 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 ∈ G and A ∈ B(G), k(x, A) denotesk(x)(A) and the probability measure
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∈Ivα i v = 1and setα = (α i v; v ∈ V, i ∈ Iv). DefineDG
α the set of all continuous functionsf : G → Rsuch that for alli ∈ I,f ◦ ei is C2on the interior of J
i 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∈I−v,Li<∞ αiv lim r→Li− (f ◦ ei)0(r) + X i∈I−v,Li=∞ αiv lim r→0−(f ◦ ei) 0(r).
Since αwill be fixed, DG
α will simply be denoted by D. When Gˆv is a star graph as
defined before, to the half lineEˆv
i, we associate the parameterαiv. Setαv= (αiv; i ∈ Iv)
andDˆv= D ˆ Gv
αv. Forf ∈ Dandx = ei(r) ∈ G\V, setf
0(x) = (f ◦e
i)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≤t such 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 ≤ u and 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)x∈GsupE[(Ku,vf (x) − Ks,tf (x)) 2] = 0;
6. For allf ∈ C0(G),x ∈ G,s ≤ t, we have lim
y→xE[(Ks,tf (y) − Ks,tf (x)) 2] = 0;
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.K1
s,t(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(EG
α)simply by (E), and we will also denote(EGˆv
αv)simply by( ˆE
v). 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(FK
s,t; 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 ˆ Fs,t:= ∨v∈VF ˆ Kv s,t ; s ≤ t
is independent on disjoint time intervals. Then, there exists a unique (up to modification) SFKKonGsuch that
• FK
s,t⊂ ˆFs,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,v s },
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 onGˆ vsuch that • for alls ≤ t,Fˆs,t:= ∨v∈VF ˆ Kv s,t ⊂ Fs,tK,
• ( ˆKv, W )is a solution of( ˆEv)for eachv ∈ V
and such that ifρx,v
s is defined by (2.1), then for alls < t inRand x ∈ Gv, a.s. on the
event{t < ρx,v
Note that (2.2) can be rewritten : for all bounded measurable functionsf onG, and allx ∈ Gv
Ks,tf (x) = ˆKs,tfˆv(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 Fˆ
s,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 α
i
vandβv = 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 ifKˆ
v1 and
ˆ
Kv2are independent givenW.
Proposition 2.5. Letv1andv2be two vertices inV such thatβv2 6= βv1 and
|βv2− 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
(FKˆv1 s,t ∨ F
ˆ Kv2
s,t )s≤t is 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 := (Wi
s,t; 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. LetK = ( ˆˆ Kv)
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,f ∈ ˆˆ Dv,x ∈ ˆˆ Gv,
a.s. ˆ Ks,tv f (ˆˆx) = ˆf (ˆx) +X i∈Iv Z t s ˆ Ks,uv (1Eˆv i ˆ f0)(ˆx)Wi(du) +1 2 Z t s ˆ Ks,uv fˆ00(ˆx)du. (3.2) Then we have the following
Lemma 3.1. For allv ∈ V,i ∈ Iv and alls ≤ t, we haveFW i s,t ⊂ F
Proof. Lety = ˆev
i(r) ∈ ˆEvi. Following Lemma 6 [8], we prove thatKˆs,tv (y) = δeˆv
i(r+Ws,ti )
for alls ≤ t ≤ σy s where
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
ˆ Fs,t:= ∨v∈VF ˆ Kv s,t; 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 • FK
s,t⊂ ˆFs,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(1Eif 0)(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 (ˆx
v). (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 eventAc
s,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 ≤ τ x s i−1v ∗ ˆKv s,t(ˆxv), if t > τsx (i.e. forA ∈ B(G), i−1v ∗ ˆKv
s,t(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 of
ˆ
Ks,tv (ˆxv)is included iniv(Gv)so
thatK0
s,t(x) ∈ P(G). Remark also that onAs,t∩ {v = ei(r + Ws,τi x s)}, a.s.
Ks,t0 (x) = i−1v ∗ ˆKs,tv (ˆxv).
Lemma 3.3. For alls < t < uand allµ ∈ P(G), a.s. onAs,u,
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) − µK
0 s,tK 0 t,u(A)|] ≤ Z G E[|Ks,u0 (x, A) − K 0 s,tK 0 t,u(x, A)|]µ(dx).
Letiandrbe such thatx = ei(r).
When t ≤ τx
s, set Y = ei(r + Ws,ti ). If u ≤ τsx, then it is easy to see that (3.6)
holds after having remarked that τY
t = τsx. If t ≤ τsx < u, then Ks,t0 (x) = δY and K0
s,u(x) = i−1v ∗ ˆKs,uv (ˆxv)withv = ei(r + Ws,τi x
s). We still haveτ Y
t = τsxwhich is now less
thanu. WriteKs,t0 Kt,u0 (x) = Kt,u0 (Y ) = i−1v0 ∗ ˆKv 0 t,u( ˆY v)wherev0 = e i(e−1i (Y ) + W i t,τY t ). Sincee−1i (Y ) = r + Wi s,t, we have v0= ei(r + Ws,ti + W i t,τY t ) = ei(r + Ws,τi x s) = v.
SinceKˆvis a flow, we get
Ks,t0 Kt,u0 (x) = i−1v ∗ ˆKt,uv ( ˆYv) = i−1v ∗ ˆKs,tv Kˆt,uv (ˆxv) = i−1v ∗ ˆKs,uv (ˆxv) = Ks,u0 (x).
Whent > τx
s, thenKs,t0 (x) = i−1v ∗ ˆKs,tv (ˆxv)andKs,u0 (x) = i−1v ∗ ˆKs,uv (ˆxv)withvdefined
as above. Letf be a bounded measurable function onG. ThenKs,u0 f (x) = ˆKs,uv fˆv(ˆxv).
SinceKˆvis a flow, we have
Ks,u0 f (x) = ˆKs,tv Kˆt,uv fˆv(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 K0
s,t(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), ˆ
Kv
t,ufˆv(ˆyv) = Kt,u0 f (y) and thus that Kˆs,tv Kˆt,uv fˆv(ˆyv) = Ks,t0 Kt,u0 f (y). This implies the
lemma.
We will say that a random kernelKis Fellerian when for alln ≥ 1and allh ∈ C0(Gn),
we haveE[K⊗nh] ∈ C0(Gn).
Lemma 3.4. For alls < t,K0
s,tis 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)) 21
Ac 0,t.
Suppose thatxbelongs to an edgeEi. Using the convergence in probabilityWτiy 0 → Wi τx 0 asy → x, we see thatP(K0 0,τ0y(y) 6= K 0 0,τx 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 0,τx 0(x) = δv}, we have lim y→xE[(K 0 0,tf (y) − K 0 0,tf (x)) 21 Cv t] = 0. Since onCv
Lemma 3.5. LetK1 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 + nK 0 s+n,s+n+2−n. . . K 0 tn−2−n,tnK 0 tn,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,K 0 s+n,s+n+2−n,. . . , K 0 tn−2−n,tn,K 0 tn,t are independent by (3.3)). DefineΩn
s,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 Ωn
s,t ⊂ Au,v. Let Ωs,t = ∪nΩns,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 + nK 0 s+n,s+n+2−n. . . K 0 tn−2−n,tnK 0 tn,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 = µK m s,tmK m tm,t+mK m t+m,t+m+2−m· · · K m um,u = µKs,tm mK m tm,tK m t,t+mK m t+m,t+m+2−m· · · K m um,u = µKs,tmKt,um = µKs,tKt,u.
This proves thatKsatisfies the flow property. Fixk ≥ 1,h ∈ C0(Gk). Let > 0andn
1∈ Nsuch thatP(ns,t> n1) < . Then for all (x, y) ∈ Gk× Gk, since
P(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 sinceKn
s,tis Fellerian for alln, we deduce that lim sup y→x |E[K ⊗k s,th(y)] − E[K ⊗k s,th(x)]| ≤ 2khk∞.
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 > 0such 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] ≤ 2kf k
2
∞P(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 whenx 6∈ V. Whenx = v ∈ V, then a.s. onAs,t,Ks,tf (x) = ˆKs,tv fˆv(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 (1Eif 0)(x)Wi(du) +1 2 Z t s Ks,u0 f00(x)du.
Proof. Letx = ei(r)withi ∈ Iv. Recall the notationxˆv = iv(x) ∈ ˆGv. 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 )1{τx s>t}+ X v∈V ˆ Ks,tv fˆv(ˆxv)1Bv s,t. Thus a.s. onAs,t, Ks,t0 f (x) = f (x) + 1{τx s>t} Z t s (f ◦ ei)0(r + Ws,ui )W i(du) +1 2 Z t s (f ◦ ei)00(r + Ws,ui )du + X v∈V 1Bv s,t X j∈Iv Z t s ˆ Ks,uv 1Eˆv j( ˆf v)0(ˆx v)Wj(du) + Z t s ˆ Ks,uv ( ˆfv)00(ˆxv)du = f (x) + 1{τx s>t} Z t s Ks,u0 (1Eif 0)(x)Wi(du) +1 2 Z t s Ks,u0 f00(x)du +X v∈V 1Bv s,t X j∈Iv Z t s Ks,u0 (1Ejf 0)(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
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 sinceΩn
s,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 nK n tn,tf (x) = Ks,tn n f +X i∈I Z t tn Ktnn,u(1Eif 0)Wi(du) +1 2 Z t tn Ktnn,uf00du ! (x) = Ks,tn nf (x) + X i∈I Z t tn Ks,tnnKtnn,u(1Eif 0)(x)Wi(du) +1 2 Z t tn Ks,tn nKtnn,uf00(x)du = f (x) +X i Z t s Ks,un (1Eif 0)(x)Wi(du) +1 2 Z t s Ks,un f00(x)du,
by independence of increments and using the fact thatKn
s,tn(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(1Eif 0)(x)Wi(du) +1 2 Z t s Ks,uf00(x)du. (3.7)
Proof. Note that for alln, onΩn
s,t, for allu ∈ [s, t], a.s. Ks,u(x) = Ks,un (x). Thus a.s. on Ωn
s,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 ≤ tand 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,v
s }(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 + mK 0 s+m,s+m+2−m. . . K 0 tm−2−m,tmK 0 tm,t(x). Clearly on{t < ρs,v s }, a.s. K 0 s,s+m(x) = i −1 v ∗ ˆK v s,s+m(ˆx
v) and for all y in the support of K0 s,s+m (x),K0 s+m,s+m+2−m (y) = i−1v ∗ ˆKv s+m,s+m+2−m (ˆyv). Thus, on{t < ρs,v s }, a.s. Ks,s0 + mK 0 s+m,s+m+2−m(x) = i −1 v ∗ ˆK v s,s+m ˆ Ksv+ m,s+m+2−m(ˆx v).
The same argument shows that on{t < ρs,v s }, a.s. Ks,t(x) = i−1v ∗ ˆK v s,s+ m ˆ Ksv+ m,s+m+2−m. . . ˆK v tm−2−m,tm ˆ Ktv m,t(ˆx v) = i−1 v ∗ ˆK v s,t(ˆx v).
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
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(1Eif 0)(x)Wi(du) +1 2 Z t s Ks,uf00(x)du.
Following [10, Lemma 3], one can prove thatFWi
s,t ⊂ Fs,tK for alli ∈ I ands ≤ t. In this
section, we will prove the following
Theorem 4.1. For eachv ∈ V, there exists a unique (up to modification) SFKKˆv on ˆ Gvsuch that • for alls ≤ t,Fˆs,t:= ∨v∈VF ˆ Kv s,t ⊂ Fs,tK,
• for allv ∈ V,( ˆKv, Wv)is a solution to the SDE : for alls ≤ t,f ∈ ˆˆ Dv,x ∈ ˆˆ Gv, a.s. ˆ Ks,tv f (ˆˆx) = ˆf (ˆx) +X i∈Iv Z t s ˆ Ks,uv (1Eˆv i ˆ f0)(ˆx)Wi(du) +1 2 Z t s ˆ Ks,uv fˆ00(ˆx)du. (4.1)
and such that defining fors ∈ Randx ∈ Gv,ρx,vs by (3.4), we have that for allt > s, a.s.
on the event{t < ρx,v
s }, (3.5) holds.
Proof. Fixv ∈ V. Fors ∈ Randx = ˆˆ ev
i(r) ∈ ˆGv withx ∈ iˆ v(Gv), setx = ei(r)(and we
havex = iˆ v(x)). Recall the definition ofτsx. Recall also the definition ofAs,tfrom (3.1).
Define the kernelKˆs,t0,v by : • OnAc s,t:Kˆ 0,v s,t(ˆx) = δxˆ; • OnAs,t: Letx = ˆˆ evi(r). Ifˆx = ˆevi(r) ∈ iv(Gv),τsx < tandei(r + Ws,τi x s) = v, define ˆ
Ks,t0,v(ˆx) = iv∗ Ks,t(ei(r)). And otherwise, defineKˆ 0,v
s,t(ˆx) = δˆev
i(r+Ws,ti ).
Now forn ≥ 1, set
ˆ Ks,tn,v= ˆK0,v s,s+ n ˆ K0,v s+ n,s+n+2−n . . . ˆKt0,v n−2−n,tn ˆ Kt0,v n,t, ifs+ n ≤ tandKˆ n,v s,t = ˆK 0,v s,t, ifs+n > t.
DefineΩns,t, Ωs,t and ns,tas in Section 3.1 and finally set Kˆs,tv = ˆK n,v
s,t, wheren = ns,t
and Kˆv
s,t(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 ∗ ˆK 0,v
s,t(ˆxv). Since for all s ≤ tandx ∈ ˆˆ Gv, a.s. onAs,t,Kˆs,tv = ˆK
0,v
s,t, 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(EG
α)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 thatPi∈Iα
byeithe orientation ofEi and letα+ =Pi∈I+α
i,α−= 1 − α+. Letα = (αi)
i∈I. In this
section, we denote(EαG)simply by(E).
The construction of flows associated to(E)is based on the skew Brownian motion (SBM) flow studied by Burdzy and Kaspi in [3]. LetW be a real white noise, then the Burdzy-Kaspi (BK) flowY associated to W andβ ∈ [−1, 1]is a SFM (see Section 7 for the definition) solution to
Ys,t(x) = x + Ws,t+ βLs,t(x), (5.1)
whereLs,t(x)is the local time ofYs,·(x)at timet. Forx ∈ G,i ∈ I andr ∈ Rsuch that x = ei(r), define τsx= inf{t ≥ s : ei(r + Ws,t) = 0}. 5.1 The caseα+6= 1 2. Fork ≥ 1, let∆k =u ∈ [0, 1]k:P k i=1ui= 1
. From [8], we recall the following Theorem 5.1. Letm+ andm− be two probability measures respectively on∆n+ and
∆n− satisfying :∀i ∈ [1, n+]andj ∈ [1, n−],
(+) Z ∆n+ uim+(du) = αi α+, (−) Z ∆n− ujm−(du) = αj+n+ α− .
(a) There exists a solution (K, W ) on G unique in law such that if Y is the BK flow associated toW andβ = 2α+− 1, then for alls ≤ tin
R,x ∈ Ga.s. (i) Ifx = ei(r), thenKs,t(x) = δei(r+Ws,t)on{t ≤ τ x s}. (ii) On {t > τx s}, Ks,t(x) is supported on {ei(Ys,t(r)), i ∈ I+} if Ys,t(r) > 0 and on {ei(Ys,t(r)), i ∈ I−}ifYs,t(r) ≤ 0. (iii) On{t > τx
s, ±Ys,t(r) > 0},Us,t±(x) = (Ks,t(x, Ei), i ∈ I±)is independent ofW and
has for lawm±.
(b) For all SFKKsuch that(K, W )solves(E), there exist two probability measuresm+
andm− respectively on ∆n+ and∆n− satisfying conditions(+) and(−)and such that
(i), (ii), (iii) above are satisfied. Let U+ = (U+(i), i ∈ I
+)and U− = (U−(j), j ∈ I−)be two random variables with
values in∆n+and∆n− such that for each(i, j) ∈ I+× I−
P(U+(i) = 1) = α i α+, P(U
−(j) = 1) = αj α−.
Note that all coordinates ofU±are equal to0expect for one coordinate which is equal therefore to1. Withm+ andm− being respectively the laws ofU+ andU−, K
s,t(x) = δϕs,t(x)whereϕis a SFM. The flowϕis also the law unique SFM solving(E).
ToU+ = (αi
α+, i ∈ I+)andU−= (α j
α−, j ∈ I−), is associated in the same way a Wiener
i.e. σ(W )-measurable solutionKW of(E)which is also the unique (up to modification)
Wiener solution to(E). 5.2 The caseα+= 12.
In this case(E)admits only one solutionKW which is Wiener, no other solutions can
be constructed by adding randomness toW. The expression ofKW is the same as the
6
Conditional independence : proof of Proposition 2.5.
In this section, we assume that for all i ∈ I, Wi = W for some real white noise
W. Our purpose is to establish Proposition 2.5 already proved in [10] in a very partic-ular case. The main idea was the following : let(ϕ+, W )and (ϕ−, −W ) be two SFM’s
solutions to Tanaka’s equation
ϕ±s,t(x) = x ± Z t
s
sgn(ϕ±s,u(x))dWu.
We know that the laws of (ϕ+, W ) and (ϕ−, W ) are unique [17]. Let ϕ = (ϕ+, ϕ−), then if(Fs,tϕ)s≤t is i.d.i, the law ofϕis unique. An intuitive explanation for this is that t 7→ |ϕ+0,t(0)| = Wt− inf0≤u≤tWu and t 7→ |ϕ−0,t(0)| = sup0≤u≤tWu− Wt do not have
common zeros after0so that sgn(ϕ+0,t(0))should be independent of sgn(ϕ−0,t(0)). In the general situation, the previous reflecting Brownian motions are replaced by two SBM’s associated toW and distinct skew parameters.
The proof of Proposition 2.5 will strongly rely on the following lemma.
Lemma 6.1. Let(β1, β2) ∈ [−1, 1]2withβ16= β2and|β2− β1| ≥ 2β1β2. Letx, y ∈ Rand
letX, Y be solutions of
Xt= x + Wt+ β1Lt(X) and Yt= y + Wt+ β2Lt(Y )
whereLt(X)andLt(Y )denote the symmetric local times at0ofX andY. Ifx 6= yor if x = y = 0, then a.s. for allt > 0,Xt6= Yt.
Proof. Assume first thatx = y = 0. It is straightforward to see that the lemma holds whenβ1≤ 0 ≤ β2. The other cases follow from [2, Theorem 1.4 (i)-(ii)].
Assume now that x 6= y and setT = inf{t > 0; Xt = Yt = 0}. Then necessarily, if T < ∞, we haveXT = YT = 0. So we can conclude using the strong Markov property
at timeT.
Proof of Proposition 2.5. To simplify the notation, fori ∈ {1, 2}, Gi, βi, Ii,Ii,± andKi
will denote respectivelyGˆvi,β
vi, Ivi,I ± vi and
ˆ
Kvi. We will also denote the edges ofGi
by(ei
j)j∈Ii and setFs,t= FK 1 s,t ∨ FK
2
s,t for alls ≤ t.
It is easy to see that ifK1andK2are independent givenW, then(F
s,t)s≤t is i.d.i.
Assume now that(Fs,t)s≤t is i.d.i. Fori ∈ {1, 2}, let Yi be the BK flow associated
toW and βi. Using the flow property, the stationarity of the flows and the fact that (Fs,t)s≤tis i.d.i., we only need to prove that for allt > 0,
K0,t1 and K0,t2 are independent givenW. (6.1) Forn ≥ 1andi ∈ {1, 2}, let(xi
j = eiki j
(ri
j), 1 ≤ j ≤ n)benpoints inGi, wherekji ∈ Iiand ri
j∈ R. Define
τji= inf{u ≥ 0 : rij+ W0,u= 0}.
Proving (6.1) reduces to prove that
(K0,t1 (x1j))1≤j≤n and (K0,t2 (x 2
j))1≤j≤nare independent givenW (6.2)
for arbitrarynand(xij).
Note that whent ≤ τji, thenK0,ti (xij)is a measurable function ofW. ForJ
1and J2
two subsets of{1, . . . , n}, let
AJ1,J2 =
which belongs toσ(W ). Then, proving (6.2) reduces to check that (for allJ1 andJ2),
givenW, onAJ1,J2,
(K0,t1 (x1j))j∈J1 and (K0,t2 (x2j))j∈J2 are independent. (6.3)
Forj ∈ Ji, define
gij= sup{u ≤ t : Y0,ui (rij) = 0}.
Note that a.s. onAJ1,J2, by Lemma 6.1,
g1 j : j ∈ J 1} ∩g2 j : j ∈ J 2} = ∅. LetJ = {Jk; 1 ≤ k ≤ m}be a partition of {1} × J1 ∪ {2} × J2
such that for allk, we haveJk⊂ {i} × Jifor somei ∈ {1, 2}and define the event
BJ = n
gij= g i0
j0 if and only if∃ ksuch that (i, j), (i0, j0) ∈ Jk× Jk o \ n ∀k < k0; if (i, j), (i0, j0) ∈ Jk× Jk0 thengij< gi 0 j0 o \ AJ1,J2.
Then a.s.{BJ}J is a partition ofAJ1,J2.
For all k, choose (ik, jk) ∈ Jk and let gk = gjikk. Let u1 < · · · < um−1 be fixed dyadic
numbers and
C := Cu1,...,um−1 = BJ ∩ {gk < uk < gk+1; 1 ≤ k ≤ m − 1}.
Letu = um−1. OnC, for all(i, j) ∈ Jm,K0,ti (xij)isFu,t∨ F0,uW-measurable. Indeed : fix (j+, j−) ∈ Ii,+× Ii,−and define
Xi,j u = ( ei j+(Y i 0,u(rji)), ifY0,ui (rji) ≥ 0; ei j−(Y v 0,u(rij)), ifY0,ui (rji) < 0. ThenXi,j u isF0,uW-measurable,
inf{r ≥ u : Ku,ri (Xui,j) = δ0} = inf{r ≥ u : Y0,ri (r i
j) = 0} ≤ gm
andK0,ti (xij) = Ku,ti (Xui,j).
Moreover, on C, for all (i, j) ∈ ∪m−1k=1Jk, K0,ti (xij) is F0,u∨ Fu,tW-measurable (since Yi
0,·(rij)does not touch0in the interval[u, t]). SinceC ∈ σ(W ),F0,u∨ Fu,tW andFu,t∨ F0,uW
are independent givenW, we deduce that onC, Ki
0,t(xvj), (i, j) ∈ Jm and Ki 0,t(xij), (i, j) ∈ ∪ m−1
k=1Jkare independent givenW. Now an immediate induction
per-mits to show that givenW, onC, (6.3) is satisfied. Since the dyadic numbersu1, · · · , um
are arbitrary, we deduce that conditionally onW, onBJ, (6.3) is satisfied and finally
givenW, onAJ1,J2, (6.3) holds.
7
Appendix 1 : The Burdzy-Kaspi flow
In this section, we show how our construction can be simplified on some particular graphs using the BK flow [3]. Let(Ws,t)s≤t be a real white noise. Forβ = ±1, the flow
associated to (5.1) has a simple expression which will be referred as the BK flow. For a fixedβ ∈] − 1, 1[, Burdzy and Kaspi constructed a SFM (see 1.7 in [3]) satisfying
(ii) With probability equal to1: for alls ∈ Rand allx ∈ R, (Ys,t(x), Ls,t(x))satisfies (5.1) and Ls,t(x) = lim ε→0+ 1 2ε Z t s
1{|Ys,u(x)|≤ε}du.
The statement (i) is a consequence of the definition ofY (see also [8, Section 3.1]) and (ii) can be found in [3, Proposition 1]. The BK flow satisfies also a strong flow property : Proposition 7.1. (1) Fixx ∈ Rand letSbe an(FW
−∞,r)r∈R-finite stopping time. Then YS,S+·(x) is the unique strong solution of the SBM equation with parameter β
driven byWS,S+·. In particularYS,S+· is independent ofF−∞,SW .
(2) LetS ≤ T be two(FW
−∞,r)r∈R-finite stopping times. Then a.s. for allu ≥ 0, x ∈ R, YS,T +u(x) = YT ,T +u◦ YS,T(x).
Proof. (1) Let Gt = F−∞,S+tW , t ≥ 0. Then YS,S+·(x) is G-adapted, WS+·− WS is a G
-Brownian motion and by (ii) above, a.s.∀t ≥ 0,
YS,S+t(x) = x + WS,S+t+ βLS,S+t(x) withLS,S+t(x) = lim ε→0+ 1 2ε Z t 0
1{|YS,S+u(x)|≤ε}du. Now (1) follows from the main result of
[11].
(2) Fixx ∈ R. Then by the previous lines for ally ∈ R, a.s.∀t ≥ 0, YT ,T +t(y) = y + WT ,T +t+ βLT ,T +t(y).
SinceYT ,T +·is independent ofF−∞,TW andYS,T isF−∞,TW measurable (by the definition
ofY), it holds that a.s.∀t ≥ 0,
YT ,T +t(YS,T(x)) = YS,T(x) + WT ,T +t+ βLT ,T +t(YS,T(x)).
Now, set
Zt= YS,t(x)1{S≤t≤T }+ YT ,t◦ YS,T(x)1{t>T }.
Then, we easily check that a.s.∀t ≥ S, Zt= x + WS,t+ β lim ε→0+ 1 2ε Z t S 1{|Zu|≤ε}du.
Since the SDE (5.1) has a unique strong solution, a.s. ∀t ≥ T,YS,t(x) = YT ,t◦ YS,T(x).
Now using (i) above, (2) holds a.s. for allx ∈ R.
Consider equation(E)defined on the graphGof Figure 4. We will sometimes iden-tifyGwith the real line. Applying Theorem 2.4, we deduce that there exists a unique (up to modification) SFMϕonRsuch that for alls ≤ tand ally ∈ R, a.s.
ϕs,t(y) = y + Ws,t+ X v∈V
(2α+v − 1)Lv,ys,t(ϕ)
whereLv,ys,t(ϕ) = lim→021 Rt
s1{|ϕs,u(y)−v|≤}du. This flow is also a Wiener solution to the
previous equation. Using the BK flow, another construction ofϕis possible : to each vertexv, let us attach the BK flowYvassociated toW, withβ
v := 2α+v − 1. Ifx = ei(r),
defineϕs,t(x) = ei(r +Ws,t)until hitting a vertex pointv1at times1, then “define”ϕs,t(x)
byYv1
s1,t(0)until hitting another vertexv2. Afters2,ϕs,t(x)will be “given by”Y v2
v
Figure 4: GraphG.
From Proposition 7.1, we deduce thatϕsolves(E), moreover ϕhas independent and stationary increments and satisfy the flow property.
In [10], it is proved that flows solutions of(E)defined on graphs like in Figure 3 have modifications which satisfy strong flow properties similar to Proposition 7.1 (2) (see [10, Corollary 2]). Actually on graphs with arbitrary orientation and transmission pa-rameters and such that each vertex has at most two adjacent edges, we can proceed to a direct construction of “global” flows using strong flow properties of “local” flows.
8
Appendix 2 : Complement to Section 5
8.1 A key lemma on BK flow
In this section, we use the same notations as in the beginning of Section 5. LetY be the BK flow (defined in the previous section) associated toW andβ := 2α+−1. For each u < v, letnbe the first integer such that]u, v[contains a dyadic number of ordernand
f (u, v)be the smallest dyadic number of orderncontained in]u, v[(f is a deterministic machinery which associates to each(u, v), u < va dyadic number in]u, v[).
For alls ≤ tand all(x, y) ∈ R2, using the conventioninf ∅ = +∞, set Tx,ys = inf{u ≥ s, Ys,u(x) = Ys,u(y)},
τs(x) = inf{u ≥ s, x + Ws,u= 0},
ns,t(x) = inf{n ≥ 1, Ys,t(x − 2−n) = Ys,t(x + 2−n)}.
For alls ≤ tand allx ∈ R, letn = ns,t(x)and define
vs,t(x) = f (s, Tx−2s −n,x+2−n)if t ≥ Tx−2s −n,x+2−n
andvs,t(x) = 0otherwise. Now let(n, v) = ns,t(x), vs,t(x)and define ys,t(x) = f Ys,v(x − 2−n), Ys,v(x + 2−n)
if t ≥ Tx−2s −n,x+2−n
andys,t(x) = 0 otherwise. Note that(s, t, x, ω) 7−→ (vs,t(x, ω), ys,t(x, ω)) is measurable
and that for alls < t,(vs,t, ys,t)isFs,tW-measurable.
Lemma 8.1. LetsandxinR. Then a.s. for allt > τs(x), we have
(i) n = ns,t(x) < ∞,
(ii) vs,t(x) = f (s, Tx−2s −n,x+2−n)and
ys,t(x) = f (Ys,v(x − 2−n), Ys,v(x + 2−n)),
(iii) Ys,t(x) = Yv,t(y), with(v, y) = (vs,t(x), ys,t(x)).
Proof. See [8, Lemma 3].
8.2 Construction of a flow of mappings
In this section, we will use the same notations as in Section 8.1 and in Section 5 with the assumptionα+6=1
2 ifn ≥ 3(nis the number of half lines constitutingG). Moreover,
we set
G+= {0} ∪ ∪i∈I+Ei, G
We will review the construction of the unique flow of mappings solving(E)defined onG. LetW be a real white noise. First we will constructϕs,·(x) for all(s, x) ∈ Q × GQ whereGQ = {z ∈ G, |z| ∈ Q+}. Denote this set of points by (si, xi)i≥0 and write xi = eji(ri)whereri ∈ Rand ji ∈ {1, · · · , n}. Letγ
+, γ− be two independent random
variables respectively taking their values inI+and in I− and such that fori ∈ I+ and j ∈ I−, P(γ+= i) = α i α+ andP(γ −= j) = αj α−.
We will construct ϕs0,·(x0), then ϕs1,·(x1) and so on. Let D be the set of all dyadic
numbers onRand{(γ+
r, γr−), r ∈ D}be a family of independent copies of(γ+, γ−)which
is also independent ofW. Ifx = ei(r), recall the definitionτsx = τs(r)whereτs(r)is as
in the previous paragraph. Forx0= ej0(r0), defineϕs0,·(x0)by
ϕs0,t(x0) = ej0(r0+ Ws0,t), ifs0≤ t ≤ τ x0 s0; 0, ift > τx0 s0, Ys0,t(r0) = 0; eh(Ys0,t(r0)), ifγ + r = h, t > τsx00, Ys0,t(r0) > 0; eh(Ys0,t(r0)), ifγ − r = h, t > τsx00, Ys0,t(r0) < 0,
wherer = f (u, v)andu, vare respectively the last zero beforetand the first zero aftert
ofYs0,·(r)(well defined whenYs0,t(r0) 6= 0). Now, suppose thatϕs0,·(x0), · · · , ϕsq−1,·(xq−1)
are defined and let{(γ+
r, γr−), r ∈ D}be a new family of independent copies of(γ+, γ−)
(that is independent of all vectors(γ+, γ−)used untilq − 1and independent also ofW).
Let
t0= infu ≥ sq : Ysq,u(rq) ∈ {Ysi,u(ri), i ∈ [0, q − 1]} .
Sincet0 < ∞, leti ∈ [0, q − 1]and (si, ri)such thatYsq,t0(rq) = Ysi,t0(ri). Now define
ϕsq,·(xq)by ϕsq,t(xq) = ejq(rq+ Wsq,t), ifsq≤ t ≤ τ xq sq; 0, ifτxq sq < t < t0, Ysq,t(rq) = 0; eh(Ys0,t(r0)), ifγ + r = h, τ xq sq < t < t0, Ysq,t(rq) > 0; eh(Ysq,t(rq)), ifγ − r = h, τ xq sq < t < t0, Ysq,t(rq) < 0; ϕsi,t(xi), ift ≥ t0,
whereris defined as inϕs0,·(x0)(from the skew Brownian motionYsq,·(rq)). In this way,
we construct(ϕsi,·(xi))i≥0.
Extension. Now we will define entirelyϕ. Lets ≤ t andx ∈ G be such that(s, x) /∈ Q × GQ. Ifx = ei(r)ands ≤ t ≤ τsx, defineϕs,t(x) = ei(r + Ws,t). Ift > τsx, letmbe the
first nonzero integer such thatYs,t(r − 2−m) = Ys,t(r + 2−m)(whenmdoes not exist we
give an arbitrary definition toϕs,t(x)). Then consider the dyadic numbers v = fs, Tr−2s −m,r+2−m
, r0= f Ys,v(r − 2−m), Ys,v(r + 2−m)
(8.1) and finally setϕs,t(x) = ϕv,t(z), where
z = e1(r0)if r0 ≥ 0and z = en++1(r0)if r0 < 0. (8.2)
Note thatϕs,t(x, ω)is measurable with respect to(s, t, x, ω).
By [8, Lemma 3], for a “typical”(s, x)a.s. for allt > τx
s,mis finite. Note also that :
for alls ≤ tand allx = ei(r) ∈ Ga.s.
This is clear when(s, x) ∈ Q × GQand remains true for alls, tandxby Lemma 8.1 (iii). The independence of the increments ofϕis clear and the stationarity comes from the fact that for all s ≤ t andx = ei(r) ∈ G(even when (s, x) ∈ Q × GQ), if v and r0 are
defined by (8.1), then on the event{t > τx
s}, a.s.ϕs,t(x) = ϕv,t(z)withzgiven by (8.2).
Writing Freidlin-Sheu formula [8, Theorem 3] for the Walsh’s Brownian motiont 7→ ϕs,s+t(x)and using (8.3), we see thatϕsolves(E).
The flow ϕis the unique SFM solving(E)in our case. Whenα+ = 1
2, the BK flow
is the trivial flowx + Ws,t which is non coalescing. The above construction cannot be
applied ifn ≥ 3, no flow of mappings solving(E)can be constructed in this case. Remark 8.2. Recall Theorem 5.1 and the lines after (the SFM case). Then(U+, U−)
can be identified with a couple(γ+, γ−)with law described as above (defineγ+ = i if
U+(i) = 1andγ− = jifU−(j) = 1). We have seen that working directly with(γ+, γ−)
instead of(U+, U−)makes the construction more clear.
8.3 The other solutions. Supposeα+6= 1
2and letm
+andm− be two probability measures as in Theorem 5.1.
Then, to(m+, m−)is associated a SFKKsolution of(E)constructed similarly toϕ. Let U+ = (U+(i))
i∈I+ and U
− = (U−(j))
j∈I− be two independent random variables with
respective values in[0, 1]n+and[0, 1]n− such that
U+ law= m+, U− law= m−. In particular a.s. P i∈I+U +(i) =P i∈I−U −(j) = 1. Let{(U+ r , Ur−), r ∈ D}be a family of
independent copies of(U+, U−)which is independent ofW. Then define
Ks0,t(x0) = δej0(r0+Ws0,t), if s0≤ t ≤ τ x0 s0; δ0, if t > τsx00, Ys0,t(r0) = 0; X i∈I+ Ur+(i)δei(Ys0,t(r0)), if t > τ x0 s0, Ys0,t(r0) > 0; X j∈I− Ur−(j)δej(Ys0,t(r0)), if t > τ x0 s0, Ys0,t(r0) < 0, whereU+ r = (Ur+(i))i∈I+, U −
r = (Ur−(j))j∈I− and r is the same as in the definition of
ϕs0,·(x0). NowKis constructed following the same steps as in the construction ofϕ.
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